#  $B$ meson anomalies within the triplet vector boson model to the light of
recent measurements from LHCb

J. M. Cabarcas<EMAIL_ADDRESS>Universidad Santo Tomás, Colombia  J.
H. Muñoz<EMAIL_ADDRESS>Departamento de Física, Universidad del Tolima,
Código Postal 730006299, Ibagué, Colombia  Néstor Quintero
<EMAIL_ADDRESS>Facultad de Ciencias Básicas, Universidad
Santiago de Cali, Campus Pampalinda, Calle 5 No. 62-00, Código Postal 76001,
Santiago de Cali, Colombia  Eduardo Rojas<EMAIL_ADDRESS>Departamento de
Física, Universidad de Nariño, A.A. 1175, San Juan de Pasto, Colombia

###### Abstract

The triplet vector boson (TVB) is a simplified new physics model involving
massive vector bosons transforming as a weak triplet vector. Such a model has
been proposed as a combined explanation of the anomalous $b\to
s\mu^{+}\mu^{-}$ and $b\to c\tau\bar{\nu}_{\tau}$ data (the so-called $B$
meson anomalies). In this work, we carry out an updated view of the TVB model
by incorporating the most recent 2022 and 2023 LHCb measurements on the lepton
flavor universality ratios $R(D^{(\ast)})={\rm BR}(B\to
D^{(\ast)}\tau\bar{\nu}_{\tau})/{\rm BR}(B\to
D^{(\ast)}\ell^{\prime}\bar{\nu}_{\ell^{\prime}})$, $R(\Lambda_{c})={\rm
BR}(\Lambda_{b}\to\Lambda_{c}\tau\bar{\nu}_{\tau})/{\rm
BR}(\Lambda_{b}\to\Lambda_{c}\mu\bar{\nu}_{\mu})$, and $R_{K^{(\ast)}}={\rm
BR}(B\to K^{(\ast)}\mu^{+}\mu^{-})/{\rm BR}(B\to K^{(\ast)}e^{+}e^{-})$. We
perform a global fit to explore the allowed parameter space by the new data
and all relevant low-energy flavor observables. Our results are confronted
with the recent high-mass dilepton searches at the Large Hadron Collider
(LHC). We find that for a heavy TVB mass of 1 TeV a common explanation of the
$B$ meson anomalies is possible for all data with the recent LHCb measurements
on $R(D^{(\ast)})$, in consistency with LHC constraints. However, this
framework is in strong tension with LHC bounds when one considers all data
along with the world average values (BABAR, Belle, and LHCb) on
$R(D^{(\ast)})$. Future measurements will be required in order to clarify such
a situation. In the end, the implications of our phenomenological analysis of
the TVB model to some known flavor parametrizations are also discussed.

##  I Introduction

In the last ten years, approximately, the high-energy physics community has
been a witness of discrepancies between experimental measurements and the
Standard Model (SM) calculations in several observables involving $b\to
s\mu^{+}\mu^{-}$ (neutral-current) and $b\to c\tau\bar{\nu}_{\tau}$ (charged-
current) transitions, which provide an important test of lepton flavor
universality (LFU). Such inconsistencies indicate strong signals of LFU
violation (for very recent interesting reviews, see Refs. London:2021lfn ;
Albrecht:2021tul ; Bifani:2018zmi ). For the neutral-current $b\to
s\mu^{+}\mu^{-}$ transition, the ratio of semileptonic decay channels,

$\displaystyle R_{K^{(\ast)}}=\frac{{\rm BR}(B\to
K^{(\ast)}\mu^{+}\mu^{-})}{{\rm BR}(B\to K^{(\ast)}e^{+}e^{-})},$ (1)

provides a test of $\mu/e$ LFU for different dilepton mass-squared range
$q^{2}$ ($q^{2}$ bins). From 2014 to 2021, the LHCb experiment reported the
existence of discrepancies between the SM predictions and the experimental
measurements (low and central $q^{2}$ bins) of $R_{K}$, $R_{K^{\ast}}$,
$R_{K_{S}}$, and $R_{K^{\ast+}}$ Aaij:2014ora ; Aaij:2019wad ; Aaij:2017vbb ;
Aaij:2021vac ; LHCb:2021lvy , hinting toward LFU violation in the $\mu/e$
sector. However, at the end of 2022, an improved LHCb analysis of the ratios
$R_{K^{(\ast)}}$, namely LHCb:2022qnv ; LHCb:2022zom

$\displaystyle R_{K}$ $\displaystyle=$
$\displaystyle\begin{cases}0.994^{+0.090+0.029}_{-0.082-0.027},\ \
q^{2}\in[0.1,1.1]\ {\rm GeV}^{2},\\\ 0.949^{+0.042+0.022}_{-0.041-0.022},\ \
q^{2}\in[1.1,6.0]\ {\rm GeV}^{2},\end{cases}$ (2)

and

$\displaystyle R_{K^{\ast}}$ $\displaystyle=$
$\displaystyle\begin{cases}0.927^{+0.093+0.036}_{-0.087-0.035},\ \
q^{2}\in[0.1,1.1]\ {\rm GeV}^{2},\\\ 1.027^{+0.072+0.027}_{-0.068-0.026},\ \
q^{2}\in[1.1,6.0]\ {\rm GeV}^{2},\end{cases}$ (3)

now shows a good agreement with the SM LHCb:2022qnv ; LHCb:2022zom . In
addition, the CMS experiment has presented a new measurement of the branching
ratio of $B_{s}\to\mu^{+}\mu^{-}$ more consistent with the SM CMS:2022mgd .
Despite that the tension on $R_{K^{(\ast)}}$ ratios and ${\rm
BR}(B_{s}\to\mu^{+}\mu^{-})$ has now disappeared, there are still some
discrepancies in the measurements of additional $b\to s\mu^{+}\mu^{-}$
observables, such as angular observables and differential branching fractions
related with $B\to K^{\ast}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$
decays Aaij:2013qta ; Aaij:2015oid ; Aaij:2020nrf ; Aaij:2013aln ;
Aaij:2015esa ; Aaij:2020ruw . Within a model-independent effective Hamiltonian
approach and under the hypothesis that New Physics (NP) couples selectively to
the muons, different scenarios with NP operators (dimension-six) have been
surveyed in the literature Aebischer:2019mlg ; Altmannshofer:2021qrr ;
Alguero:2021anc ; Alguero:2019ptt ; Geng:2021nhg ; Hurth:2021nsi ;
Angelescu:2021lln ; Carvunis:2021jga ; London:2021lfn ; Greljo:2022jac ;
Alguero:2023jeh . The most recent global fit analysis Greljo:2022jac ;
Alguero:2023jeh  taking into account updated $b\to s\mu^{+}\mu^{-}$ data
(including $R_{K^{(\ast)}}$ by LHCb LHCb:2022qnv ; LHCb:2022zom  and ${\rm
BR}(B_{s}\to\mu^{+}\mu^{-})$ by CMS CMS:2022mgd ), showed that the Wilson
coefficient (WC) solution $C^{bs\mu\mu}_{9}=-C^{bs\mu\mu}_{10}$, related with
the operators $(\bar{s}P_{L}\gamma_{\alpha}b)(\bar{\mu}\gamma^{\alpha}\mu)$
and $(\bar{s}P_{L}\gamma_{\alpha}b)(\bar{\mu}\gamma^{\alpha}\gamma_{5}\mu)$,
is still a viable solution to describe the data.

On the other hand, the experimental measurements collected by the BABAR,
Belle, and LHCb experiments on different charged-current $b\to
c\tau\bar{\nu}_{\tau}$ observables, indicate the existence of disagreement
with respect to the SM predictions Lees:2012xj ; Lees:2013uzd ;
Huschle:2015rga ; Sato:2016svk ; Hirose:2017vbz ; Aaij:2015yra ; Aaij:2017deq
; Aaij:2017uff ; Belle:2019rba ; Hirose:2017dxl ; Hirose:2016wfn ;
Abdesselam:2019wbt ; HFLAV:2022pwe ; LHCb2022 ; LHCb:2023zxo ; LHCb2023 ;
HFLAVsummer ; Aaij:2017tyk  (see Table 1 for a summary). Regarding the ratios
of semileptonic $B$ meson decays,

$R(D^{(\ast)})=\dfrac{{\rm BR}(B\to D^{(\ast)}\tau\bar{\nu}_{\tau})}{{\rm
BR}(B\to D^{(\ast)}\ell^{\prime}\bar{\nu}_{\ell^{\prime}})},$ (4)

with $\ell^{\prime}=e\ {\rm or}\ \mu$ (the so-called $R(D^{(\ast)})$
anomalies), the LHCb has presented, very recently, the first combined
measurement using Run 1 data (3 fb-1) with muonic $\tau$ decay reconstruction
LHCb2022 ; LHCb:2023zxo ,

$\displaystyle R(D)_{\rm LHCb22}$ $\displaystyle=$ $\displaystyle 0.441\pm
0.060\pm 0.066,$ (5) $\displaystyle R(D^{\ast})_{\rm LHCb22}$ $\displaystyle=$
$\displaystyle 0.281\pm 0.018\pm 0.024,$ (6)

which show a tension of $1.9\sigma$ with the SM predictions. Additionally, the
LHCb also reported a preliminary measurement of $R(D^{\ast})$ using partial
Run 2 data (2 fb-1), where the $\tau$ is hadronically reconstructed LHCb2023 .
When combined with Run 1, the result is LHCb2023

$R(D^{\ast})_{\rm LHCb23}=0.257\pm 0.012\pm 0.018,$ (7)

that is compatible with SM at the $\sim 1\sigma$ level. Incorporating these
new LHCb results, the preliminary world average values reported by the Heavy
Flavor Averaging Group (HFLAV) are HFLAVsummer

$\displaystyle R(D)_{\rm HFLAV23}$ $\displaystyle=$ $\displaystyle 0.356\pm
0.029,$ (8) $\displaystyle R(D^{\ast})_{\rm HFLAV23}$ $\displaystyle=$
$\displaystyle 0.284\pm 0.013,$ (9)

that now exceed the SM by $3.2\sigma$. Moreover, the LHCb measurement of the
ratio $R(J/\psi)={\rm BR}(B_{c}\to J/\psi\tau\bar{\nu}_{\tau})/{\rm
BR}(B_{c}\to J/\psi\mu\bar{\nu}_{\mu})$ Aaij:2017tyk  also shows tension
($\sim 2\sigma$) with regard to the SM prediction Harrison:2020nrv .
Additional hints of LFU violation in the $b\to c\tau\bar{\nu}_{\tau}$
transition have been obtained in the Belle measurements of the $\tau$ lepton
polarization $P_{\tau}(D^{\ast})$ Hirose:2017dxl ; Hirose:2016wfn  and the
longitudinal polarization of the $D^{*}$ meson $F_{L}(D^{\ast})$
Abdesselam:2019wbt  related with the channel $\bar{B}\to
D^{\ast}\tau\bar{\nu}_{\tau}$, which also exhibit a deviation from the SM
values Iguro:2022yzr . The tauonic channel $B_{c}\to
J/\psi\tau\bar{\nu}_{\tau}$ has not been measured yet, but indirect
constraints on its branching ratio have been imposed $<30\%$ Alonso:2016oyd
and $<10\%$ Akeroyd:2017mhr . In Table 1 we summarize the current experimental
measurements and their corresponding SM predictions. We also collect in Table
1 the experimental and theoretical values of the ratio of inclusive decays
$R(X_{c})\equiv{\rm BR}(B\to X_{c}\tau\bar{\nu}_{\tau})/{\rm BR}(B\to
X_{c}\mu\bar{\nu}_{\mu})$, which is generated via the same $b\to
c\tau\bar{\nu}_{\tau}$ transition Kamali:2018bdp . The SM estimation on
$R(X_{c})$ is based on the $1S$ mass scheme and includes nonperturbative
corrections of the order $\mathcal{O}(1/m_{b}^{3})$, while the NP effects took
into account the subleading $\mathcal{O}(1/m_{b})$ corrections Kamali:2018bdp
. The $R(D^{(\ast)})$ anomalies still exhibit the largest deviation. The other
$b\to c\tau\bar{\nu}_{\tau}$ observables also show tension (moderate) with the
data, although, some of them have large experimental uncertainties (such as
$R(J/\psi)$ and $P_{\tau}(D^{\ast})$). While the ratio $R(X_{c})$ is in
excellent agreement with the SM.

In addition, the LHCb Collaboration has recently released the first
measurement of the ratio of semileptonic $\Lambda_{b}$ baryon decays, namely
LHCb:2022piu

$R(\Lambda_{c})\equiv\dfrac{{\rm
BR}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\tau^{-}\bar{\nu}_{\tau})}{{\rm
BR}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\bar{\nu}_{\mu})}=0.242\pm 0.076,$
(10)

in agreement at the $\sim 1.2\sigma$ level with the most recent SM
calculation, $R(\Lambda_{c})_{\rm SM}=0.324\pm 0.004$ Bernlochner:2018bfn . In
Eq. (10) we have added in quadrature the statistical and systematic
uncertainties, and the external branching ratio uncertainty from the channel
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\bar{\nu}_{\mu})$ LHCb:2022piu . It
is interesting to highlight that this new measurement is below the SM value,
pointing to an opposite direction than the current $b\to
c\tau\bar{\nu}_{\tau}$ data (see Table 1). Nevertheless, in order to provide
an overall picture, all the anomalous $b\to c\tau\bar{\nu}_{\tau}$ data must
be taken into account. To the best of our knowledge, the impact of the new
LHCb measurement on $R(\Lambda_{c})$ has been recently studied from a model-
independent way (effective field theory approach) Fedele:2022iib  and in the
singlet vector leptoquark model Garcia-Duque:2022tti .

Table 1: Experimental status and SM predictions on observables related to the charged-current transitions $b\to c\ell\bar{\nu}_{\ell}$ ($\ell=\mu,\tau$). Transition | Observable | Expt. measurement | SM prediction
---|---|---|---
$b\to c\tau\bar{\nu}_{\tau}$ | $R(D)$ |  $0.441\pm 0.060\pm 0.066$ (LHCb22) LHCb2022 ; LHCb:2023zxo  | 0.298 $\pm$ 0.004 HFLAVsummer
|  |  $0.356\pm 0.029$ (HFLAV) HFLAVsummer  |
| $R(D^{\ast})$ |  $0.281\pm 0.018\pm 0.024$ (LHCb22) LHCb2022 ; LHCb:2023zxo  | 0.254 $\pm$ 0.005 HFLAVsummer
|  |  $0.257\pm 0.012\pm 0.018$ (LHCb23) LHCb2023  |
|  |  $0.284\pm 0.013$ (HFLAV) HFLAVsummer  |
| $R(J/\psi)$ |  $0.71\pm 0.17\pm 0.18$ Aaij:2017tyk  | 0.2582 $\pm$ 0.0038 Harrison:2020nrv
| $P_{\tau}(D^{\ast})$ |  $-0.38\pm 0.51^{+0.21}_{-0.16}$ Hirose:2017dxl ; Hirose:2016wfn  |  $-0.497\pm 0.007$ Iguro:2022yzr
| $F_{L}(D^{\ast})$ |  $0.60\pm 0.08\pm 0.035$ Abdesselam:2019wbt  |  $0.464\pm 0.003$ Iguro:2022yzr
| $R(X_{c})$ | 0.223 $\pm$ 0.030 Kamali:2018bdp  | 0.216 $\pm$ 0.003 Kamali:2018bdp
| ${\rm BR}(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})$ |  $<10\%$ Akeroyd:2017mhr , $<30\%$ Alonso:2016oyd  |  $(2.16\pm 0.16)\%$ Gomez:2019xfw
$b\to c\mu\bar{\nu}_{\mu}$ | $R_{D}^{\mu/e}$ |  $0.995\pm 0.022\pm 0.039$ Glattauer:2015teq  |  $0.9960\pm 0.0002$ Becirevic:2020rzi
| $R_{D^{\ast}}^{\mu/e}$ |  $0.961\pm 0.050$ Belle:2017rcc  |  $0.9974\pm 0.0001$ Bobeth:2021lya

Although the $b\to c\tau\bar{\nu}_{\tau}$ data is suggesting stronger signals
of LFU violation than $b\to s\mu^{+}\mu^{-}$ one, a combined explanation of
the current data is still desirable. This simultaneous explanation can be
generated by different tree-level heavy mediators with adequate couplings, for
example, charged scalar bosons, extra gauge bosons or leptoquarks (scalar and
vector). For an extensive list of literature, see the theoretical status
report presented in Ref. London:2021lfn . In this work, we will pay particular
attention to the common explanation provides by the so-called Triplet Vector
Boson (TVB) model Calibbi:2015kma ; Bhattacharya:2014wla ; Greljo:2015mma ;
Faroughy:2016osc ; Buttazzo:2017ixm ; Bhattacharya:2016mcc ; Kumar:2018kmr ;
Guadagnoli:2018ojc ; Boucenna:2016wpr ; Boucenna:2016qad  111Let us notice
that in a recent work Capdevila:2020rrl , the TVB model was implemented as an
explanation to the Cabibbo angle anomaly and $b\to s\ell^{+}\ell^{-}$ data.,
in which the SM is extended by including a color-neutral real $SU(2)_{L}$
triplet of massive vectors $W^{\prime}$ and $Z^{\prime}$ that coupled
predominantly to left-handed (LH) fermions from the second- and third-
generations Calibbi:2015kma ; Bhattacharya:2014wla ; Greljo:2015mma ;
Faroughy:2016osc ; Buttazzo:2017ixm ; Bhattacharya:2016mcc ; Kumar:2018kmr ;
Guadagnoli:2018ojc ; Boucenna:2016wpr ; Boucenna:2016qad . The neutral boson
$Z^{\prime}$ is responsible for the $b\to s\mu^{+}\mu^{-}$ data, while the
charged boson $W^{\prime}$ generates the $b\to c\tau\bar{\nu}_{\tau}$ one. We
adopt a phenomenological approach of the TVB model based on the minimal setup
of couplings between the new gauge bosons $Z^{\prime},W^{\prime}$ and LH
fermions of the SM, without specifying the complete UV model. We present an
updated analysis of TVB model (parametric space) by including the new 2022 and
2023 LHCb data on $R_{K^{(\ast)}}$, $R(D^{(\ast)})$, and $R(\Lambda_{c})$. We
also incorporate in our study all relevant flavor observables that are also
affected by this NP model, such as $B_{s}-\bar{B}_{s}$ mixing, neutrino
trident production, LFV decays ($B\to K^{(\ast)}\mu^{\pm}\tau^{\mp}$,
$B_{s}\to\mu^{\pm}\tau^{\mp}$, $\tau\to\mu\phi$,
$\Upsilon(nS)\to\mu^{\pm}\tau^{\mp}$), rare $B$ decays ($B\to
K^{(\ast)}\nu\bar{\nu},B\to K\tau^{+}\tau^{-},B_{s}\to\tau^{+}\tau^{-}$), and
bottomonium LFU ratios. Furthermore, we study the consistency of the allowed
TVB parameter space with the Large Hadron Collider (LHC) bounds from searches
of high-mass dilepton resonances at the ATLAS experiment.

Even though our focus will be phenomenological, regarding the ultra violet
(UV) complete realization for the TVB model, the extension of the SM must
allow for Lepton Flavor Non Universal (LFNU) couplings to the extra gauge
bosons and LFV. In this direction in Ref. Boucenna:2016qad  there is a
proposal in which an extra $SU(2)$ gauge group is added and where extra
scalars, new vector-like fermions and some non trivial transformations under
the SM group are included. It is clear, that the couplings of fermions to the
extra gauge bosons of the particular UV realization, will have model-dependent
consequences that might relate different terms between them; however, since we
make emphasis that our approach is phenomenological, we will start from the
most general lagrangian for the TVB model as possible, and we will make
comparisons to other approaches presented in Refs. Calibbi:2015kma ;
Greljo:2015mma ; Buttazzo:2017ixm ; Bhattacharya:2016mcc  where the new
physics is coupled predominantly to the second and third generation of left
handed quarks and leptons, ensuring LFNU and LFV through different mechanisms.
Restrict our results to a particular UV-model is out of our target.

This paper is structured as follows: in Sec. II we discuss the main aspects of
the TVB model to accommodate the $B$ meson anomalies. As a next step in Sec.
III, we consider the most relevant flavor observables and present the TVB
model contributions to them. The LHC bounds are also studied. We then perform
our phenomenological analysis of the allowed parametric space in Sec. IV and
our conclusions are presented in Sec. V.

##  II The Triplet Vector boson model

In general, flavor anomalies have been boarded into the current literature as
a motivation to build innovative models and to test well established New
Physics (NP) models. In this section, we focus in the previously mentioned
Triplet Vector Boson (TVB) model Bhattacharya:2014wla ; Calibbi:2015kma ;
Kumar:2018kmr ; Faroughy:2016osc ; Greljo:2015mma ; Bhattacharya:2016mcc ;
Buttazzo:2017ixm ; Guadagnoli:2018ojc ; Boucenna:2016wpr ; Boucenna:2016qad
as a possible explanations of these anomalies, that might accommodate the
observed flavor experimental results. One significant feature of this model,
is the inclusion of extra SM-like vector bosons with non-zero couplings to the
SM fermions, that allow us to include additional interactions.

In the fermion mass basis, the most general lagrangian describing the dynamics
of the fields can be written as

$\displaystyle\Delta{\cal
L}_{V}=g_{ij}^{q}(\bar{\Psi}_{iL}^{Q}\gamma^{\mu}\sigma^{I}\Psi^{Q}_{jL})V^{I}_{\mu}+g_{ij}^{\ell}(\bar{\Psi}_{iL}^{\ell}\gamma^{\mu}\sigma^{I}\Psi^{\ell}_{jL})V^{I}_{\mu}$
(11)

where, $V_{\mu}$ stands for the extra or new vector bosons that transform as
(1, 3, 0) under the $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ gauge
symmetry and must be redefined as $W^{\prime\pm},\,Z^{\prime}$. On the other
side, SM fermions are arranged into the doublets $\Psi^{Q}_{L}$ and
$\Psi^{\ell}_{L}$ given by

$\displaystyle\Psi^{Q}_{L}=\begin{pmatrix}V^{\dagger}u_{L}\cr
d_{L}\end{pmatrix},\qquad\Psi^{\ell}_{L}=\begin{pmatrix}\nu_{L}\cr\ell_{L}\end{pmatrix}.$
(12)

It is worth noticing here that in this particular model the CKM mixing matrix
$V$ is applied on the up-type quarks.

In order to find the effective lagrangian for this model, the heavy degrees of
freedom corresponding to vector bosons introduced above must be integrated
out. Introducing the definition for the currents
$J_{Q}=\bar{\Psi}_{iL}^{Q}\gamma^{\mu}\sigma^{I}\Psi^{Q}_{jL}$ and
$J_{\ell}=\bar{\Psi}_{iL}^{\ell}\gamma^{\mu}\sigma^{I}\Psi^{\ell}_{jL}$, the
effective lagrangian is therefore

$\displaystyle{\cal L}_{eff}$ $\displaystyle=$
$\displaystyle-\frac{(g_{ij}^{q}J_{Q}+g_{ij}^{\ell}J_{\ell})^{2}}{2M^{2}_{V}}$
(13) $\displaystyle=$
$\displaystyle-\frac{(g_{ij}^{q}J_{Q})^{2}}{2M^{2}_{V}}-\frac{g_{ij}^{q}g_{kl}^{\ell}J_{Q}J_{\ell}}{M^{2}_{V}}-\frac{(g_{ij}^{\ell}J_{\ell})^{2}}{2M^{2}_{V}}.$
(14)

The middle term of the right-hand side of the above equation corresponds to

$\displaystyle\frac{g_{ij}^{q}g_{kl}^{\ell}J_{Q}J_{\ell}}{M^{2}_{V}}$
$\displaystyle=$
$\displaystyle\frac{g_{ij}^{q}g_{kl}^{\ell}}{M_{V}^{2}}(\bar{\Psi}^{Q}_{iL}\gamma_{\mu}\sigma^{I}\Psi^{Q}_{jL})(\bar{\Psi}^{\ell}_{kL}\gamma^{\mu}\sigma^{I}\Psi^{\ell}_{lL})$
(15)

Substituting equation (12) in the last expression, it leads us to

$\displaystyle\frac{g_{ij}^{q}g_{kl}^{\ell}J_{Q}J_{\ell}}{M^{2}_{V}}$
$\displaystyle=$ $\displaystyle
2\frac{g_{kl}^{\ell}}{M_{V}^{2}}\left[(Vg^{d})_{ij}(\,\bar{u}_{iL}\gamma_{\mu}d_{jL})(\bar{\ell}_{k}\gamma^{\mu}\nu_{lL})+{\rm
h.c.}\right]$ (16)
$\displaystyle+\frac{g_{kl}^{\ell}}{M_{V}^{2}}\left[(Vg^{d}V^{\dagger})_{ij}(\bar{u}_{iL}\gamma_{\mu}u_{jL})(\bar{\nu}_{kL}\gamma^{\mu}\nu_{lL})+g_{ij}^{d}(\bar{d}_{iL}\gamma_{\mu}d_{jL})(\bar{\ell}_{kL}\gamma^{\mu}\ell_{lL})\right]$
$\displaystyle-\frac{g_{kl}^{\ell}}{M_{V}^{2}}\left[(Vg^{d}V^{\dagger})_{ij}(\bar{u}_{iL}\gamma_{\mu}u_{jL})(\bar{\ell}_{kL}\gamma^{\mu}\ell_{lL})+g_{ij}^{d}(\bar{d}_{iL}\gamma_{\mu}d_{jL})(\bar{\nu}_{kL}\gamma^{\mu}\nu_{lL})\right],$

in this expression, we can identify that the first term expresses an effective
interaction of the SM fields that should be mediated by extra bosonic charged
fields, while the remaining terms are mediated by an extra neutral bosonic
field. These mediators are precisely the vector boson fields ${W^{\prime}}$
and ${Z^{\prime}}$ introduced in this model and which masses can naively be
considered to be (almost) degenerated which is required by electroweak
precision data Faroughy:2016osc . For simplicity, and without losing
generality, we are going to consider that the couplings $g^{q,\ell}$ are real
to avoid CP violation effects. Additionally, it is important to notice that we
can write compactly the couplings of quarks to the vector boson fields with an
explicit dependence in the couplings of the down sector and also, keeping in
mind that the CKM matrix couples into the doublets to up-type quarks and that
we should restrict the significant contributions for the second and third
families. For this purpose, we restrict the relevant couplings of the down
sector to $g_{bb}$, $g_{ss}$ and $g_{sb}=g_{bs}$ while other terms remain
zero. This hypothesis that the couplings to the first generation of fermions
(also in the leptonic sector) can be neglected has been widely accepted in the
literature into the context of flavor anomaly explanations Calibbi:2015kma ;
Kumar:2018kmr ; Greljo:2015mma ; Faroughy:2016osc ; Bhattacharya:2016mcc ;
Bhattacharya:2014wla ; Buttazzo:2017ixm ; Guadagnoli:2018ojc . Lastly, the
resultant compact form for the couplings of the quark sector to the
$W^{\prime}$ that we obtained are

$\displaystyle g_{\alpha b}$ $\displaystyle=$ $\displaystyle g_{bb}V_{\alpha
b}+g_{sb}V_{\alpha s},$ $\displaystyle g_{\alpha s}$ $\displaystyle=$
$\displaystyle g_{ss}V_{\alpha s}+g_{sb}V_{\alpha b},$ (17)

where $\alpha$ stands for $u,c$ or $t$ quark flavors. The same procedure
described above must be implemented for a compact form of the couplings of up-
type quarks to the $Z^{\prime}$ boson. In this case we find two possibilities:
one on flavor conserving interaction given by

$\displaystyle g_{\alpha\alpha}$ $\displaystyle=$ $\displaystyle
g_{bb}V_{\alpha b}^{2}+2g_{\alpha b}V_{\alpha s}V_{\alpha b}+g_{ss}V_{\alpha
s}^{2};$ (18)

the other is related to flavor changing $Z^{\prime}$ couplings mediated by

$\displaystyle g_{\alpha\beta}=g_{bb}V_{\beta b}V_{\alpha b}+g_{sb}V_{\beta
s}V_{\alpha b}+g_{sb}V_{\beta b}V_{\alpha s}+g_{ss}V_{\beta s}V_{\alpha s},$
(19)

where $\alpha\neq\beta$ labels $u,c$ or $t$ quark flavors.

To close this kind of parametrization, we mention that the terms of the r.h.s
of equation (15) are responsible for and will be important to $4q$ and $4\ell$
interactions ruled by the lagrangian

$\displaystyle{\cal
L}_{NP}^{4q,4\ell}=-\frac{g_{ij}^{q}g_{kl}^{q}}{2M_{V}^{2}}(\bar{\Psi}^{Q}_{iL}\gamma_{\mu}\sigma^{I}\Psi^{Q}_{jL})(\bar{\Psi}^{Q}_{kL}\gamma^{\mu}\sigma^{I}\Psi^{Q}_{lL})-\frac{g_{ij}^{\ell}g_{kl}^{\ell}}{2M_{V}^{2}}(\bar{\Psi}^{\ell}_{iL}\gamma_{\mu}\sigma^{I}\Psi^{\ell}_{jL})(\bar{\Psi}^{\ell}_{kL}\gamma^{\mu}\sigma^{I}\Psi^{\ell}_{lL})$
(20)

###  II.1 Other parametrizations

In this subsection, we compare the previous parameterization explained above
with others used in some representative references studied widely in the TVB
model.

In the TVB model presented in refs Calibbi:2015kma ; Bhattacharya:2016mcc ,
the mixing pattern for quarks is enriched by the inclusion of mixing matrices
that will rotate the fields from the gauge basis to the mass basis and a
projector ($X,Y$) that will ensure the dominance of the second and third
families to explain anomalies. Particularly, the explicit form of these
matrices for the down-type quarks and charged leptons and projectors are

$\displaystyle D=\begin{pmatrix}1&0&0\cr 0&\cos\theta_{D}&\sin\theta_{D}\cr
0&-\sin\theta_{D}&\cos\theta_{D}\end{pmatrix},\qquad L=\begin{pmatrix}1&0&0\cr
0&\cos\theta_{L}&\sin\theta_{L}\cr
0&-\sin\theta_{L}&\cos\theta_{L}\end{pmatrix},\qquad
X=Y=\begin{pmatrix}0&0&0\cr 0&0&0\cr 0&0&1\end{pmatrix}.$ (21)

These matrices will leave an explicit dependence of these mixing angles
($\theta_{D,L}$) into the couplings to the extra fields, which by the
experimental results coming from different observables, can be constrained.
The assumptions made in the introduction of these matrices were previously
introduced in Calibbi:2015kma , and we can establish the full equivalence
between the notations of the angles by the relations $\theta_{D}=\alpha_{sb}$
and $\theta_{L}=\alpha_{\mu\tau}$. We also found that these couplings can be
translated to the generic parameterization introduced at the beginning of this
section. For this purpose, as it was explained before, the couplings of all
the quark sector will be dependent on the couplings of the down-type quarks,
particularly in this kind of parameterization, we can illustrate the way that
the couplings are obtained through the effective charged lagrangian that will
be given as

$\displaystyle{\cal L}_{\rm
eff}^{W^{\prime}}=2\frac{g_{2}^{q}g_{2}^{\ell}}{M_{V}^{2}}\left[(V\,D^{\dagger}X\,D)_{ij}(\,\bar{u}_{iL}\gamma_{\mu}d_{jL})(L^{\dagger}Y\,L)_{kl}(\bar{\ell}_{k}\gamma^{\mu}\nu_{lL})+{\rm
h.c}\right];$ (22)

thus, we obtain the equivalence

$\displaystyle g_{bb}$ $\displaystyle\to$ $\displaystyle
g_{2}^{q}\cos^{2}\theta_{D}$ $\displaystyle g_{sb}$ $\displaystyle\to$
$\displaystyle-g_{2}^{q}\sin\theta_{D}\cos\theta_{D}$ $\displaystyle g_{ss}$
$\displaystyle\to$ $\displaystyle g_{2}^{q}\sin^{2}\theta_{D},$ (23)

and for the leptonic sector

$\displaystyle g_{\tau\tau}$ $\displaystyle\to$ $\displaystyle
g_{2}^{\ell}\cos^{2}\theta_{L}$ $\displaystyle g_{\mu\tau}$ $\displaystyle\to$
$\displaystyle-g_{2}^{\ell}\sin\theta_{L}\cos\theta_{L}$ $\displaystyle
g_{\mu\mu}$ $\displaystyle\to$ $\displaystyle g_{2}^{\ell}\sin^{2}\theta_{L}.$
(24)

The comparison and equivalence among parameterizations of different
influential references can be found in Tables 2, 3, 4 and 5.

For our last comparison, we considered the parameterization given in Refs.
Greljo:2015mma ; Buttazzo:2017ixm  where the couplings to the vector bosons
have almost the same structure of the initial parameterization presented here,
but its major difference consists in the dependence on flavor matrices denoted
by the authors as $\lambda_{ij}^{(q,\ell)}$. This incidence of the flavor
structure into the model can be shown using the charged effective lagrangian
as we did before

$\displaystyle{\cal L}_{\rm
eff}^{W^{\prime}}=\frac{g_{q}g_{\ell}}{2M_{V}^{2}}\left[(V\,\lambda)_{ij}(\,\bar{u}_{iL}\gamma_{\mu}d_{jL})(\bar{\ell}_{k}\gamma^{\mu}\nu_{lL})+{\rm
h.c}\right],$ (25)

to obtain the desired dominance of couplings to the second and third families
using the flavor matrices mentioned before, the $\lambda_{ij}$ belonging to
the first family must be set to zero. Additionally, the values for
$\lambda_{bb}=\lambda_{\tau\tau}=1$ in order to maximize its contribution.
However, as an illustration, we can make a complete relation of the
implementation of the flavor matrices to the construction of couplings for the
quark sector without any assumption in Tables 2, 3, 4 and 5.

Table 2: Couplings to $W^{\prime}$ boson in different parameterizations of the TVB model Coupling | Parameterization in Kumar:2018kmr  | Parameterization in Calibbi:2015kma ; Bhattacharya:2016mcc  | Parameterization in Greljo:2015mma ; Buttazzo:2017ixm
---|---|---|---
$g_{ub}^{q}$ | $g_{bb}V_{ub}+g_{sb}V_{us}$ | $g_{2}^{q}(V_{ub}\cos^{2}\theta_{d}-V_{us}\cos\theta_{d}\sin\theta_{d})$ | $g_{q}(V_{ub}+V_{ud}\lambda_{db}+V_{us}\lambda_{sb})/\sqrt{2}$
$g_{cb}^{q}$ | $g_{bb}V_{cb}+g_{sb}V_{cs}$ | $g_{2}^{q}(V_{cb}\cos^{2}\theta_{d}-V_{cs}\cos\theta_{d}\sin\theta_{d})$ | $g_{q}(V_{cb}+V_{cd}\lambda_{db}+V_{cs}\lambda_{sb})/\sqrt{2}$
$g_{tb}^{q}$ | $g_{bb}V_{tb}+g_{sb}V_{ts}$ | $g_{2}^{q}(V_{tb}\cos^{2}\theta_{d}-V_{ts}\cos\theta_{d}\sin\theta_{d})$ | $g_{q}(V_{tb}+V_{ud}\lambda_{tb}+V_{us}\lambda_{sb})/\sqrt{2}$
$g_{us}^{q}$ | $g_{ss}V_{us}+g_{sb}V_{ub}$ | $g_{2}^{q}(V_{us}\sin^{2}\theta_{d}-V_{ub}\cos\theta_{d}\sin\theta_{d})$ | $g_{q}(V_{ud}\lambda_{ds}+V_{ub}\lambda_{sb}+V_{us}\lambda_{ss})/\sqrt{2}$
$g_{cs}^{q}$ | $g_{ss}V_{cs}+g_{sb}V_{ucb}$ | $g_{2}^{q}(V_{cs}\sin^{2}\theta_{d}-V_{cb}\cos\theta_{d}\sin\theta_{d})$ | $g_{q}(V_{cd}\lambda_{ds}+V_{cb}\lambda_{sb}+V_{cs}\lambda_{ss})/\sqrt{2}$
$g_{ts}^{q}$ | $g_{ss}V_{ts}+g_{sb}V_{tb}$ | $g_{2}^{q}(V_{ts}\sin^{2}\theta_{d}-V_{tb}\cos\theta_{d}\sin\theta_{d})$ | $g_{q}(V_{td}\lambda_{ds}+V_{tb}\lambda_{sb}+V_{ts}\lambda_{ss})/\sqrt{2}$
Table 3:  Flavor conserving couplings to $Z^{\prime}$ boson in different parameterizations of the TVB model. Coupling | Parameterization in Kumar:2018kmr  | Parameterization in Calibbi:2015kma ; Bhattacharya:2016mcc  | Parameterization in Greljo:2015mma ; Buttazzo:2017ixm
---|---|---|---
$g_{uu}^{q}$ | $g_{bb}V_{ub}^{2}+2g_{sb}V_{us}V_{ub}+g_{ss}V_{us}^{2}$ | $g_{2}^{q}(V_{ub}^{2}\cos^{2}\theta_{d}-2V_{us}V_{ub}\cos\theta_{d}\sin\theta_{d}+V_{us}^{2}\sin^{2}\theta_{d})$ | $g_{q}\lambda_{uu}/\sqrt{2}$
$g_{cc}^{q}$ | $g_{bb}V_{cb}^{2}+2g_{sb}V_{cs}V_{cb}+g_{ss}V_{cs}^{2}$ | $g_{2}^{q}(V_{cb}^{2}\cos^{2}\theta_{d}-2V_{cs}V_{cb}\cos\theta_{d}\sin\theta_{d}+V_{cs}^{2}\sin^{2}\theta_{d})$ | $g_{q}\lambda_{cc}/\sqrt{2}$
$g_{tt}^{q}$ | $g_{bb}V_{tb}^{2}+2g_{sb}V_{ts}V_{tb}+g_{ss}V_{ts}^{2}$ | $g_{2}^{q}(V_{tb}^{2}\cos^{2}\theta_{d}-2V_{ts}V_{tb}\cos\theta_{d}\sin\theta_{d}+V_{ts}^{2}\sin^{2}\theta_{d})$ | $g_{q}\lambda_{tt}/\sqrt{2}$
Table 4: Flavor changing couplings to $Z^{\prime}$ boson in different parameterizations of the TVB model. Coupling | Parameterization in Kumar:2018kmr  | Parameterization in Calibbi:2015kma ; Bhattacharya:2016mcc  | Parameterization in Greljo:2015mma ; Buttazzo:2017ixm
---|---|---|---
$g_{uc}^{q}$ | $g_{bb}V_{cb}V_{ub}+g_{sb}V_{cs}V_{ub}$ | $g_{2}^{q}V_{cb}V_{ub}\cos^{2}\theta_{d}-g_{2}^{q}V_{cs}V_{ub}\cos\theta_{d}\sin\theta_{d}$ | $g_{q}\lambda_{uc}/\sqrt{2}$
| $+g_{sb}V_{cb}V_{us}+g_{ss}V_{cs}V_{us}$ | $-g_{2}^{q}V_{cb}V_{us}\cos\theta_{d}\sin\theta_{d}+g_{2}^{q}V_{cs}V_{us}\sin^{2}\theta_{d}$ |
$g_{ut}^{q}$ | $g_{bb}V_{tb}V_{ub}+g_{sb}V_{ts}V_{ub}$ | $g_{2}^{q}V_{tb}V_{ub}\cos^{2}\theta_{d}-g_{2}^{q}V_{ts}V_{ub}\cos\theta_{d}\sin\theta_{d}$ | $g_{q}\lambda_{ut}/\sqrt{2}$
| $+g_{sb}V_{tb}V_{us}+g_{ss}V_{ts}V_{us}$ | $-g_{2}^{q}V_{tb}V_{us}\cos\theta_{d}\sin\theta_{d}+g_{2}^{q}V_{ts}V_{us}\sin^{2}\theta_{d}$ |
$g_{ct}^{q}$ | $g_{bb}V_{cb}V_{tb}+g_{sb}V_{cs}V_{tb}$ | $g_{2}^{q}V_{cb}V_{tb}\cos^{2}\theta_{d}-g_{2}^{q}V_{cs}V_{tb}\cos\theta_{d}\sin\theta_{d}$ | $g_{q}\lambda_{ct}/\sqrt{2}$
| $+g_{sb}V_{cb}V_{ts}+g_{ss}V_{cs}V_{ts}$ | $-g_{2}^{q}V_{cb}V_{ts}\cos\theta_{d}\sin\theta_{d}+g_{2}^{q}V_{cs}V_{ts}\sin^{2}\theta_{d}$ |
Table 5: Couplings of leptons to $Z^{\prime}$ boson in different parameterizations of the TVB model. Coupling | Parameterization in Kumar:2018kmr  | Parameterization in Calibbi:2015kma ; Bhattacharya:2016mcc  | Parameterization in Greljo:2015mma ; Buttazzo:2017ixm
---|---|---|---
$g_{\mu\mu}$ | $g_{\mu\mu}$ | $g_{2}^{\ell}\sin^{2}\theta_{L}$ | $g_{q}(\lambda_{\mu\mu})/\sqrt{2}$
$g_{\mu\tau}$ | $g_{\mu\tau}$ | $-g_{2}^{\ell}\sin\theta_{L}\cos\theta_{L}$ | $2g_{q}(\lambda_{\mu\tau})/\sqrt{2}$
$g_{\tau\tau}$ | $g_{\tau\tau}$ | $g_{2}^{\ell}\cos^{2}\theta_{L}$ | $g_{q}/\sqrt{2}$

We make emphasis that the results presented in tables 2, 3, 4, and 5 allow us
to understand the differences and similarities for the parameterizations
presented above in the context of the TVB model; additionally it gives us a
complete interpretation of the variables present on each one and the
possibilities to find adjustments to explain flavor anomalies.

##  III Relevant Observables

In this section, we discuss the constraints from the most relevant flavor
observables on the TVB model couplings that simultaneously accommodate the $B$
meson anomalies. We will include the recent experimental progress from Belle
and LHCb on different LFV decays (such as
$\Upsilon(1S)\to\mu^{\pm}\tau^{\mp}$, $B\to K^{\ast}\mu^{\pm}\tau^{\mp}$, and
$\tau\to\mu\phi$).

###  III.1 $b\to c\ell^{-}\bar{\nu}_{\ell}$ ($\ell=\mu,\tau$) data

The $W^{\prime}$ boson leads to additional tree-level contribution to $b\to
c\ell^{-}\bar{\nu}_{\ell}$ transitions involving leptons from second- and
third-generation $(\ell=\mu,\tau)$. The total low-energy effective Lagrangian
has the following form Gomez:2019xfw

$\displaystyle-\mathcal{L}_{\rm eff}(b\to c\ell\bar{\nu}_{\ell})_{\rm
SM+W^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}V_{cb}\Big{[}(1+C_{V}^{bc\ell\nu_{\ell}})(\bar{c}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}P_{L}\nu_{\ell})\Big{]},$
(26)

where $G_{F}$ is the Fermi coupling constant, $V_{cb}$, is the charm-bottom
Cabbibo-Kobayashi-Maskawa (CKM) matrix element, and $C_{V}^{bc\ell\nu_{\ell}}$
is the Wilson coefficient (WC) associated with the NP vector (left-left)
operator. This WC is defined as

$\displaystyle C_{V}^{bc\ell\nu_{\ell}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{4G_{F}V_{cb}}\frac{2(V_{cs}g^{q}_{sb}+V_{cb}g^{q}_{bb})g^{\ell}_{\ell\ell}}{M_{V}^{2}}\
\ \ (\ell=\mu,\tau),$ (27)

with $M_{V}$ the heavy boson mass. The NP effects on the LFU ratios $R(X)$
($X=D,D^{\ast},J/\psi$), the $D^{\ast}$ and $\tau$ longitudinal polarizations
related with the channel $\bar{B}\to D^{\ast}\tau\bar{\nu}_{\tau}$, the ratio
of inclusive decays $R(X_{c})$, and the tauonic decay
$B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau}$ can be easily parametrized as
Gomez:2019xfw

$\displaystyle R(X)$ $\displaystyle=$ $\displaystyle R(X)_{\rm
SM}\big{|}1+C_{V}^{bc\tau\nu_{\tau}}\big{|}^{2},$ (28) $\displaystyle
F_{L}(D^{*})$ $\displaystyle=$ $\displaystyle F_{L}(D^{*})_{\rm SM}\
r_{D^{\ast}}^{-1}\big{|}1+C_{V}^{bc\tau\nu_{\tau}}\big{|}^{2},$ (29)
$\displaystyle P_{\tau}(D^{*})$ $\displaystyle=$ $\displaystyle
P_{\tau}(D^{*})_{\rm SM}\
r_{D^{\ast}}^{-1}\big{|}1+C_{V}^{bc\tau\nu_{\tau}}\big{|}^{2}\ ,$ (30)
$\displaystyle R(X_{c})$ $\displaystyle=$ $\displaystyle R(X_{c})_{\rm
SM}\Big{(}1+2.294\ {\rm
Re}(C_{V}^{bc\tau\nu_{\tau}})+1.147\big{|}C_{V}^{bc\tau\nu_{\tau}}\big{|}^{2}\Big{)},$
(31) $\displaystyle{\rm BR}(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})$
$\displaystyle=$ $\displaystyle{\rm
BR}(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})_{\text{SM}}\big{|}1+C_{V}^{bc\tau\nu_{\tau}}\big{|}^{2},$
(32)

respectively, where $r_{D^{\ast}}=R(D^{*})/R(D^{*})_{\rm SM}$. For ${\rm
BR}(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})$, we will use the bound $<10\%$
Akeroyd:2017mhr . Concerning to the ratio $R(\Lambda_{c})$ very recently
measured by LHCb LHCb:2022piu , the SM contribution is also rescaled by the
overall factor $\big{|}1+C_{V}^{bc\tau\nu_{\tau}}\big{|}^{2}$, namely
Datta:2017aue

$R(\Lambda_{c})=R(\Lambda_{c})_{\rm SM}\
\big{|}1+C_{V}^{bc\tau\nu_{\tau}}\big{|}^{2}.$ (33)

A long term integrated luminosity of $50\ {\rm ab}^{-1}$ is expected to be
accumulated by the Belle II experiment Belle-II:2018jsg , allowing
improvements at the level of $\sim 3\%$ and $\sim 2\%$ for the statistical and
systematic uncertainties of $R(D)$ and $R(D^{\ast})$, respectively Belle-
II:2018jsg . It is also envisioned accuracy improvements on angular analysis
in $\bar{B}\to D^{\ast}\tau\bar{\nu}_{\tau}$ decay ($\tau$ polarization
observable $P_{\tau}(D^{*})$), as well as on $q^{2}$-distribution Belle-
II:2018jsg . On the other hand, the LHCb will be able to improve measurements
of $R(D^{\ast})$ and $R(J/\psi)$ in the future runs of data taking
Albrecht:2021tul ; Bifani:2018zmi .

In regard to the transition $b\to c\mu\bar{\nu}_{\mu}$, the $\mu/e$ LFU ratios
$R_{D^{(\ast)}}^{\mu/e}\equiv{\rm BR}(B\to D^{(\ast)}\mu\bar{\nu}_{\mu})/{\rm
BR}(B\to D^{(\ast)}e\bar{\nu}_{e})$ have to be taken into account. The
experimental values obtained by Belle Glattauer:2015teq ; Belle:2017rcc  are
in great accordance with the SM estimations Becirevic:2020rzi ; Bobeth:2021lya
(see Table 1). The $W^{\prime}$ boson coupling to lepton pair
$\mu\bar{\nu}_{\mu}$ modifies this ratio as

$R_{D^{(\ast)}}^{\mu/e}=[R_{D^{(\ast)}}^{\mu/e}]_{\rm
SM}\big{|}1+C_{V}^{bc\mu\nu_{\mu}}\big{|}^{2},$ (34)

where $C_{V}^{bc\mu\nu_{\mu}}$ is given by Eq. (27). From this LFU ratio we
get the bound

$\frac{|(V_{cs}g^{q}_{sb}+V_{cb}g^{q}_{bb})g^{\ell}_{\mu\mu}|}{M_{V}^{2}}\leqslant
0.013\ {\rm TeV}^{-2},$ (35)

which is relevant for the couplings aiming to explain the $b\to
s\mu^{+}\mu^{-}$ anomaly (see Sec. III.3).

###  III.2 $b\to u\ell^{-}\bar{\nu}_{\ell}$ ($\ell=\mu,\tau$) data

The TVB model can also induce NP contributions in the leptonic decay
$B\to\ell\bar{\nu}_{\ell}$ induced via the charged-current transition $b\to
u\ell^{-}\bar{\nu}_{\ell}$ ($\ell=\mu,\tau$). The ratio

$R_{B}^{\tau/\mu}\equiv\dfrac{{\rm BR}(B^{-}\to\tau^{-}\bar{\nu}_{\tau})}{{\rm
BR}(B^{-}\to\mu^{-}\bar{\nu}_{\mu})},$ (36)

provides a clean LFU test Becirevic:2020rzi . Through this ratio the
uncertainties on the decay constant $f_{B}$ and CKM element $V_{ub}$ cancel
out (circumventing the tension between the exclusive and inclusive values of
$V_{ub}$ UTfit:2022hsi ). The NP effects on this ratio can be expressed as

$R_{B}^{\tau/\mu}=[R_{B}^{\tau/\mu}]_{\rm
SM}\Bigg{|}\dfrac{1+C_{V}^{bu\tau\nu_{\tau}}}{1+C_{V}^{bu\mu\nu_{\mu}}}\Bigg{|}^{2},$
(37)

where

$C_{V}^{bu\ell\nu_{\ell}}=\frac{\sqrt{2}}{4G_{F}M_{U_{1}}^{2}}\Big{[}|x_{L}^{b\ell}|^{2}+\frac{V_{us}}{V_{ub}}x_{L}^{s\tau}(x_{L}^{b\ell})^{\ast}\Big{]},\
\ (\ell=\mu,\tau)$ (38)

and

$[R_{B}^{\tau/\mu}]_{\rm
SM}=\Big{(}\dfrac{m_{\tau}}{m_{\mu}}\Big{)}^{2}\Big{(}\dfrac{m_{B}^{2}-m_{\tau}^{2}}{m_{B}^{2}-m_{\mu}^{2}}\Big{)}^{2}=222.5\pm
3.0.$ (39)

The experimental value is $[R_{B}^{\tau/\mu}]_{\rm Exp}=205.7\pm 96.6$, which
was obtained from the values reported by the Particle Data Group (PDG) on
${\rm BR}(B^{-}\to\tau^{-}\bar{\nu}_{\tau})$ PDG2020  and the Belle experiment
on ${\rm BR}(B^{-}\to\mu^{-}\bar{\nu}_{\mu})$ Belle:2019iji .

###  III.3 $b\to s\mu^{+}\mu^{-}$ data

The NP effective Lagrangian responsible for the semileptonic transition $b\to
s\mu^{+}\mu^{-}$ can be expressed as

$\mathcal{L}(b\to s\mu^{+}\mu^{-})_{\rm
NP}=\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{\ast}(C^{bs\mu\mu}_{9}\mathcal{O}^{bs\mu\mu}_{9}+C^{bs\mu\mu}_{10}\mathcal{O}^{bs\mu\mu}_{10})+\
{\rm h.c.},$ (40)

where the NP is encoded in the WCs $C^{bs\mu\mu}_{9}$ and $C^{bs\mu\mu}_{10}$
of the four-fermion operators

$\displaystyle\mathcal{O}^{bs\mu\mu}_{9}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\rm
em}}{4\pi}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\mu}\gamma^{\mu}\mu),$ (41)
$\displaystyle\mathcal{O}^{bs\mu\mu}_{10}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\rm
em}}{4\pi}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\mu}\gamma^{\mu}\gamma_{5}\mu),$
(42)

respectively, with $\alpha_{\rm em}$ being the fine-constant structure. A
global fit analysis including most current $b\to s\mu^{+}\mu^{-}$ data, such
as $R_{K^{(\ast)}}$ by LHCb LHCb:2022qnv ; LHCb:2022zom  and ${\rm
BR}(B_{s}\to\mu^{+}\mu^{-})$ by CMS CMS:2022mgd , has been recently performed
in Ref. Greljo:2022jac ; Alguero:2023jeh . Among the different NP scenarios,
the $C^{bs\mu\mu}_{9}=-C^{bs\mu\mu}_{10}$ solution is preferred by the data
Greljo:2022jac ; Alguero:2023jeh .222Let us notice that the single WC
$C^{bs\mu\mu}_{9}$ also provides a good fit of the $b\to s\mu^{+}\mu^{-}$ data
Greljo:2022jac ; Alguero:2023jeh . Some explicit model examples are shown in
Greljo:2022jac . The best fit $1\sigma$ solution is Greljo:2022jac

$C^{bs\mu\mu}_{9}=-C^{bs\mu\mu}_{10}\in[-0.23,-0.11].$ (43)

In the context of the TVB model, the $Z^{\prime}$ boson induces a tree-level
contribution to $b\to s\mu^{+}\mu^{-}$ transition via the WCs

$C^{bs\mu\mu}_{9}=-C^{bs\mu\mu}_{10}=-\frac{\pi}{\sqrt{2}G_{F}\alpha_{\rm
em}V_{tb}V_{ts}^{\ast}}\frac{g_{sb}^{q}g_{\mu\mu}^{\ell}}{M_{V}^{2}}.$ (44)

Using the result of the global fit, Eq. (43), this corresponds to

$-\frac{g_{sb}^{q}g_{\mu\mu}^{\ell}}{M_{V}^{2}}\in[1.7,3.5]\times 10^{-4}\
{\rm TeV}^{-2}.$ (45)

###  III.4 Bottomonium processes: $R_{\Upsilon(nS)}$ and
$\Upsilon(nS)\to\mu^{\pm}\tau^{\mp}$

Test of LFU has also been studied in the leptonic ratio $R_{\Upsilon(nS)}$
(with $n=1,2,3$) in connection with the reported hints of LFU violation in the
charged-current transition $b\to c\tau\bar{\nu}_{\tau}$ Aloni:2017eny ;
Garcia-Duque:2021qmg .333Recently, in Ref. Descotes-Genon:2021uez  has been
proposed a new method to test LFU through inclusive dileptonic $\Upsilon(4S)$
decays. It is known that NP scenarios aiming to provide an explanation to the
anomalous $b\to c\tau\bar{\nu}_{\tau}$ data, also induce effects in the
neutral-current transition $b\bar{b}\to\tau^{+}\tau^{-}$ Aloni:2017eny ;
Garcia-Duque:2021qmg . Experimentally, the BABAR and CLEO Collaborations have
reported the values delAmoSanchez:2010bt ; Besson:2006gj ; Lees:2020kom

$\displaystyle R_{\Upsilon(1S)}$ $\displaystyle=$
$\displaystyle\begin{cases}\text{BABAR-10:}\ 1.005\pm 0.013\pm
0.022\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{delAmoSanchez:2010bt}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.9924\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aloni:2017eny}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(46) $\displaystyle R_{\Upsilon(2S)}$ $\displaystyle=$
$\displaystyle\begin{cases}\text{CLEO-07:}\ 1.04\pm 0.04\pm
0.05\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Besson:2006gj}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.9940\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aloni:2017eny}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(47) $\displaystyle R_{\Upsilon(3S)}$ $\displaystyle=$
$\displaystyle\begin{cases}\text{CLEO-07:}\ 1.05\pm 0.08\pm
0.05\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Besson:2006gj}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{BABAR-20:}\ 0.966\pm 0.008\pm
0.014\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Lees:2020kom}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.9948\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aloni:2017eny}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(48)

where the theoretical uncertainty is typically of the order
$\pm\mathcal{O}(10^{-5})$ Aloni:2017eny . These measurements are in good
accordance with the SM estimations, except for the 2020 measurement on
$R_{\Upsilon(3S)}$ that shows an agreement at the $1.8\sigma$ level
Lees:2020kom . By averaging the CLEO-07 Besson:2006gj  and BABAR-20
Lees:2020kom  measurements we obtain $R_{\Upsilon(3S)}^{\rm Ave}=0.968\pm
0.016$, which deviates at the $1.7\sigma$ level with respect to the SM
prediction Garcia-Duque:2021qmg .

The NP effects of the TVB model on the leptonic ratio can be expressed as
Aloni:2017eny ; Garcia-Duque:2021qmg

$R_{\Upsilon(nS)}=\frac{(1-4x_{\tau}^{2})^{1/2}}{|A_{V}^{\rm
SM}|^{2}}\Big{[}|A_{V}^{b\tau}|^{2}(1+2x_{\tau}^{2})+|B_{V}^{b\tau}|^{2}(1-4x_{\tau}^{2})\Big{]},$
(49)

with $x_{\tau}=m_{\tau}/m_{\Upsilon(nS)}$, $|A_{V}^{\rm SM}|=-4\pi\alpha
Q_{b}$, and

$\displaystyle A_{V}^{b\tau}$ $\displaystyle=$ $\displaystyle-4\pi\alpha
Q_{b}+\frac{m_{\Upsilon(nS)}^{2}}{4}\frac{g^{q}_{bb}g^{\ell}_{\tau\tau}}{4M_{V}^{2}},$
(50) $\displaystyle B_{V}^{b\tau}$ $\displaystyle=$
$\displaystyle-\frac{m_{\Upsilon(nS)}^{2}}{2}\frac{g^{q}_{bb}g^{\ell}_{\tau\tau}}{4M_{V}^{2}}.$
(51)

The neutral gauge boson also generates the LFV processes
$\Upsilon\to\mu^{\pm}\tau^{\mp}$ $(\Upsilon\equiv\Upsilon(nS))$. The branching
fraction is given by Bhattacharya:2016mcc ; Kumar:2018kmr

${\rm
BR}(\Upsilon\to\mu^{\pm}\tau^{\mp})=\frac{f_{\Upsilon}^{2}m_{\Upsilon}^{3}}{48\pi\Gamma_{\Upsilon}}\Big{(}2+\frac{m_{\tau}^{2}}{m_{\Upsilon}^{2}}\Big{)}\Big{(}1-\frac{m_{\tau}^{2}}{m_{\Upsilon}^{2}}\Big{)}^{2}\Big{|}\dfrac{g^{q}_{bb}(g^{\ell}_{\mu\tau})^{\ast}}{M_{V}^{2}}\Big{|}^{2},$
(52)

where $f_{\Upsilon}$ and $m_{\Upsilon}$ are the Upsilon decay constant and
mass, respectively. The decay constant values can be extracted from the
experimental branching ratio measurements of the processes $\Upsilon\to
e^{-}e^{+}$. Using current data from PDG PDG2020 , one obtains
$f_{\Upsilon(1S)}=(659\pm 17)\ {\rm MeV}$, $f_{\Upsilon(2S)}=(468\pm 27)\ {\rm
MeV}$, and $f_{\Upsilon(3S)}=(405\pm 26)\ {\rm MeV}$. Experimentally, the
reported ULs are ${\rm BR}(\Upsilon(1S)\to\mu^{\pm}\tau^{\mp})<2.7\times
10^{-6}$ from Belle Belle:2022cce , and ${\rm
BR}(\Upsilon(2S)\to\mu^{\pm}\tau^{\mp})<3.3\times 10^{-6}$, ${\rm
BR}(\Upsilon(3S)\to\mu^{\pm}\tau^{\mp})<3.1\times 10^{-6}$ from PDG PDG2020 .
From these ULs we get

$\displaystyle\Upsilon(1S)\to\mu^{\pm}\tau^{\mp}$ $\displaystyle:$
$\displaystyle\ \frac{|g^{q}_{bb}(g^{\ell}_{\mu\tau})^{\ast}|}{M_{V}^{2}}<5.7\
{\rm TeV}^{-2},$ (53a) $\displaystyle\Upsilon(2S)\to\mu^{\pm}\tau^{\mp}$
$\displaystyle:$ $\displaystyle\
\frac{|g^{q}_{bb}(g^{\ell}_{\tau\mu})^{\ast}|}{M_{V}^{2}}<6.2\ {\rm
TeV}^{-2},$ (53b) $\displaystyle\Upsilon(3S)\to\mu^{\pm}\tau^{\mp}$
$\displaystyle:$ $\displaystyle\
\frac{|g^{q}_{bb}(g^{\ell}_{\mu\tau})^{\ast}|}{M_{V}^{2}}<5.2\ {\rm
TeV}^{-2}.$ (53c)

###  III.5 $\Delta F=2$ processes: $B_{s}-\bar{B}_{s}$ and
$D^{0}-\bar{D}^{0}$mixing

The interactions of a $Z^{\prime}$ boson to quarks $s\bar{b}$ relevant for
$b\to s\mu^{+}\mu^{-}$ processes also generate a contribution to
$B_{s}-\bar{B}_{s}$ mixing DiLuzio:2019jyq ; DiLuzio:2017fdq . The NP effects
to the $B_{s}-\bar{B}_{s}$ mixing can be described by the effective Lagrangian

$\mathcal{L}_{\rm\Delta
B=2}^{Z^{\prime}}=-\frac{4G_{F}}{\sqrt{2}}|V_{tb}V_{ts}^{\ast}|^{2}C_{sb}^{LL}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{s}\gamma^{\mu}P_{L}b)+\
{\rm h.c.},$ (54)

where

$C_{sb}^{LL}=\frac{1}{4\sqrt{2}G_{F}|V_{tb}V_{ts}^{\ast}|^{2}}\frac{|g_{sb}^{q}|^{2}}{M_{Z^{\prime}}^{2}}.$
(55)

Thus, the NP contributions to the mass difference $\Delta M_{s}$ of the
neutral $B_{s}$ meson can be expressed as DiLuzio:2019jyq

$\dfrac{\Delta M_{s}^{\rm SM+NP}}{\Delta M_{s}^{\rm
SM}}=\Big{(}1+\frac{\eta^{6/23}}{R^{\rm loop}_{\rm SM}}C_{sb}^{LL}\Big{)},$
(56)

where $\eta=\alpha_{s}(M_{Z^{\prime}})/\alpha_{s}(m_{b})$ accounts for running
from the $M_{Z^{\prime}}$ scale down to the $b$-quark mass scale and the SM
loop function is $R^{\rm loop}_{\rm SM}=(1.310\pm 0.010)\times 10^{-3}$
DiLuzio:2019jyq . At present, $\Delta M_{s}$ has been experimentally measured
with great precision $\Delta M_{s}^{\rm Exp}=(17.757\pm 0.021)\ {\rm ps}^{-1}$
DiLuzio:2019jyq ; HFLAV:2022pwe . On the theoretical side, the average is
$\Delta M_{s}^{\rm SM}=(18.4^{+0.7}_{-1.2})\ {\rm ps}^{-1}$ implying that
$\Delta M_{s}^{\rm SM}/\Delta M_{s}^{\rm Exp}=1.04^{+0.04}_{-0.07}$
DiLuzio:2019jyq . This value yields to

$0.89\leq\Bigg{|}1+\frac{\eta^{6/23}}{R_{\rm loop}^{\rm
SM}}C_{sb}^{LL}\Bigg{|}\leq 1.11,$ (57)

where in the TVB model translates into the important $2\sigma$ bound

$\frac{|g_{sb}^{q}|}{M_{V}}\geq 3.9\times 10^{-3}\ {\rm TeV^{-1}}.$ (58)

In addition, the $Z^{\prime}$ boson can also admit $c\to u$ transitions,
consequently generating tree-level effects on $D^{0}-\bar{D}^{0}$ mixing
Kumar:2018kmr ; Alok:2021pdh . The effective Lagrangian describing the
$Z^{\prime}$ contribution to $D^{0}-\bar{D}^{0}$ mixing can be expressed as
Kumar:2018kmr ; Alok:2021pdh

$\mathcal{L}_{\rm\Delta
C=2}^{Z^{\prime}}=-\frac{|g_{uc}|^{2}}{2M_{Z^{\prime}}^{2}}(\bar{c}\gamma_{\mu}P_{L}u)(\bar{c}\gamma^{\mu}P_{L}u)+\
{\rm h.c.},$ (59)

where
$g_{uc}=g^{q}_{bb}V_{cb}V^{\ast}_{ub}+g^{q}_{sb}(V_{cs}V^{\ast}_{ub}+V_{cb}V^{\ast}_{us})+g^{q}_{ss}V_{cs}V^{\ast}_{us}$
Kumar:2018kmr  (see also Table 4). Such a NP contributions are constrained by
the results of the mass difference $\Delta M_{D}$ of neutral $D$ mesons. The
theoretical determination of this mass difference is limited by our
understanding of the short and long-distance contributions Kumar:2018kmr ;
Alok:2021pdh . Here we follow the recent analysis of Ref. Kumar:2018kmr
focused on short-distance SM contribution that sets the conservative (strong)
bound

$\frac{|g_{ss}^{q}|}{M_{V}}\leq 3\times 10^{-3}\ {\rm TeV^{-1}}.$ (60)

The couplings $g^{q}_{bb}$ and $g^{q}_{sb}$ are less constrained by $\Delta
M_{D}$ Kumar:2018kmr , therefore, we will skip them in our study.

###  III.6 Neutrino Trident Production

The $Z^{\prime}$ couplings to leptons from second-generation
($g_{\mu\mu}=g_{\nu_{\mu}\nu_{\mu}}$) also generate a contribution to the
cross-section of neutrino trident production (NTP),
$\nu_{\mu}N\to\nu_{\mu}N\mu^{+}\mu^{-}$ Altmannshofer:2014pba . The cross-
section is given by Altmannshofer:2014pba

$\frac{\sigma_{\rm SM+NP}}{\sigma_{\rm
SM}}=\frac{1}{1+(1+4s_{W}^{2})^{2}}\Big{[}\Big{(}1+\frac{v^{2}g_{\mu\mu}^{2}}{M_{V}^{2}}\Big{)}^{2}+\Big{(}1+4s_{W}^{2}+\frac{v^{2}g_{\mu\mu}^{2}}{M_{V}^{2}}\Big{)}^{2}\Big{]},$
(61)

where $v=(\sqrt{2}G_{F})^{-1/2}$ and $s_{W}\equiv\sin\theta_{W}$ (with
$\theta_{W}$ the Weinberg angle). The existing CCFR trident measurement
$\sigma_{\rm CCFR}/\sigma_{\rm SM}=0.82\pm 0.28$ provides the upper bound

$\frac{|g_{\mu\mu}^{\ell}|}{M_{Z^{\prime}}}\leq 1.13\ {\rm TeV^{-1}}.$ (62)

###  III.7 LFV $B$ decays: $B\to K^{(\ast)}\mu^{\pm}\tau^{\mp}$ and
$B_{s}\to\mu^{\pm}\tau^{\mp}$

The $Z^{\prime}$ boson mediates LFV transitions $b\to s\mu^{\pm}\tau^{\mp}$
($B\to K^{(\ast)}\mu^{\pm}\tau^{\mp}$ and $B_{s}^{0}\to\mu^{\pm}\tau^{\mp}$)
at tree level via the WCs Calibbi:2015kma

$C^{bs\mu\tau}_{9}=-C^{bs\mu\tau}_{10}=-\frac{\pi}{\sqrt{2}G_{F}\alpha_{\rm
em}V_{tb}V_{ts}^{\ast}}\frac{g^{q}_{sb}(g^{\ell}_{\mu\tau})^{\ast}}{M_{V}^{2}}.$
(63)

The current experimental limits ($90\%$ C.L.) on the branching ratios of
$B^{+}\to K^{+}\mu^{\pm}\tau^{\mp}$ are PDG2020

$\displaystyle{\rm BR}(B^{+}\to K^{+}\mu^{+}\tau^{-})_{\rm exp}$
$\displaystyle<$ $\displaystyle 4.5\times 10^{-5},$ (64) $\displaystyle{\rm
BR}(B^{+}\to K^{+}\mu^{-}\tau^{+})_{\rm exp}$ $\displaystyle<$ $\displaystyle
2.8\times 10^{-5}.$ (65)

Let us notice that LHCb Collaboration obtained a limit of ${\rm BR}(B^{+}\to
K^{+}\mu^{-}\tau^{+})_{\rm LHCb}<3.9\times 10^{-5}$ Aaij:2020mqb  that is
comparable with the one quoted above from PDG. On the other hand, the LHCb has
recently presented the first search of $B^{0}\to K^{\ast
0}\mu^{\pm}\tau^{\mp}$ LHCb:2022wrs . The obtained UL on this LFV decay is
LHCb:2022wrs

${\rm BR}(B^{0}\to K^{\ast 0}\mu^{\pm}\tau^{\mp})_{\rm exp}<1.0\times
10^{-5}.$ (66)

From the theoretical side, the branching ratio of $B^{+}\to
K^{+}\mu^{+}\tau^{-}$ Parrott:2022zte  and $B^{0}\to K^{\ast
0}\mu^{+}\tau^{-}$ Calibbi:2015kma  can be written as

$\displaystyle{\rm BR}(B^{+}\to K^{+}\mu^{+}\tau^{-})$ $\displaystyle=$
$\displaystyle\big{(}a_{K}|C^{bs\mu\tau}_{9}|^{2}+b_{K}|C^{bs\mu\tau}_{10}|^{2}\big{)}\times
10^{-9},$ (67) $\displaystyle{\rm BR}(B^{0}\to K^{\ast 0}\mu^{+}\tau^{-})$
$\displaystyle=$
$\displaystyle\Big{(}(a_{K^{\ast}}+c_{K^{\ast}})|C^{bs\mu\tau}_{9}|^{2}+(b_{K^{\ast}}+d_{K^{\ast}})|C^{bs\mu\tau}_{10}|^{2}\Big{)}\times
10^{-9},$ (68)

respectively, where $(a_{K},b_{K})=(12.72\pm 0.81,13.21\pm 0.81)$
Parrott:2022zte , and
$(a_{K^{\ast}},b_{K^{\ast}},c_{K^{\ast}},d_{K^{\ast}})=(3.0\pm 0.8,2.7\pm
0.7,16.4\pm 2.1,15.4\pm 1.9)$ Calibbi:2015kma  are the numerical coefficients
that have been calculated using the $B\to K^{(\ast)}$ transitions form factors
obtained from lattice QCD Parrott:2022zte ; Calibbi:2015kma . The decay
channel with final state $\mu^{-}\tau^{+}$ can be easily obtained by replacing
$\mu\leftrightarrows\tau$. The current ULs can be translated into the bounds

$\displaystyle B^{+}\to K^{+}\mu^{+}\tau^{-}$ $\displaystyle:$ $\displaystyle\
\frac{|g^{q}_{sb}(g^{\ell}_{\mu\tau})^{\ast}|}{M_{V}^{2}}<6.2\times 10^{-2}\
{\rm TeV}^{-2},$ (69a) $\displaystyle B^{+}\to K^{+}\mu^{-}\tau^{+}$
$\displaystyle:$ $\displaystyle\
\frac{|g^{q}_{sb}(g^{\ell}_{\tau\mu})^{\ast}|}{M_{V}^{2}}<4.9\times 10^{-2}\
{\rm TeV}^{-2},$ (69b) $\displaystyle B^{0}\to K^{\ast 0}\mu^{+}\tau^{-}$
$\displaystyle:$ $\displaystyle\
\frac{|g^{q}_{sb}(g^{\ell}_{\mu\tau})^{\ast}|}{M_{V}^{2}}<2.5\times 10^{-2}\
{\rm TeV}^{-2}.$ (69c)

As for the LFV leptonic decay $B_{s}\to\mu^{\pm}\tau^{\mp}$, the branching
ratio is Calibbi:2015kma

$\displaystyle{\rm BR}(B_{s}^{0}\to\mu^{\pm}\tau^{\mp})$ $\displaystyle=$
$\displaystyle\tau_{B_{s}}\frac{f_{B_{s}}^{2}m_{B_{s}}m^{2}_{\tau}}{32\pi^{3}}\alpha^{2}G_{F}^{2}|V_{tb}V_{ts}^{\ast}|^{2}\Big{(}1-\frac{m_{\tau}^{2}}{m_{B_{s}}^{2}}\Big{)}^{2}\big{(}|C^{bs\mu\tau}_{9}|^{2}+|C^{bs\mu\tau}_{10}|^{2}\big{)},$
(70)

where $f_{B_{s}}=(230.3\pm 1.3)$ MeV is the $B_{s}$ decay constant
HFLAV:2022pwe  and we have used the limit $m_{\tau}\gg m_{\mu}$. Recently, the
LHCb experiment has reported the first upper limit of ${\rm
BR}(B_{s}\to\mu^{\pm}\tau^{\mp})<4.2\times 10^{-5}$ at $95\%$ CL Aaij:2019okb
. Thus, one gets the following limit

$\frac{|g^{q}_{sb}(g^{\ell}_{\mu\tau})^{\ast}|}{M_{V}^{2}}<5.1\times 10^{-2}\
{\rm TeV}^{-2}.$ (71)

###  III.8 Rare $B$ decays: $B\to K^{(\ast)}\nu\bar{\nu}$, $B\to
K\tau^{+}\tau^{-}$ and $B_{s}\to\tau^{+}\tau^{-}$

Recently, the interplay between the di-neutrino channel $B\to
K^{(\ast)}\nu\bar{\nu}$ and the $B$ meson anomalies has been studied by
several works Alok:2021pdh ; Bause:2020auq ; Bause:2021cna ; Browder:2021hbl ;
He:2021yoz . In the NP scenario under study, the $Z^{\prime}$ boson can give
rise to $B\to K^{(\ast)}\nu\bar{\nu}$ at tree level. The effective Hamiltonian
for the $b\to s\nu\bar{\nu}$ transition is given by Buras:2014fpa

$\mathcal{H}_{\rm eff}(b\to s\nu\bar{\nu})=-\frac{\alpha_{\rm
em}G_{F}}{\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}C_{L}^{ij}(\bar{s}\gamma^{\mu}P_{L}b)(\bar{\nu}_{i}\gamma_{\mu}(1-\gamma_{5})\nu_{j}),$
(72)

where $C_{L}^{ij}=C_{L}^{\rm SM}+\Delta C_{L}^{ij}$ is the aggregate of the SM
contribution $C_{L}^{\rm SM}\approx-6.4$ and the NP effects $\Delta
C_{L}^{ij}$, that in the TVB framework read as

$\Delta C_{L}^{ij}=\frac{\pi}{\sqrt{2}G_{F}\alpha_{\rm
em}V_{tb}V_{ts}^{\ast}}\frac{g^{q}_{sb}g^{\ell}_{ij}}{M_{V}^{2}},$ (73)

with $i,j=\mu,\tau$. By defining the ratio Buras:2014fpa

$R^{\nu\bar{\nu}}_{K^{(\ast)}}\equiv\frac{{\rm BR}(B\to
K^{(\ast)}\nu\bar{\nu})}{{\rm BR}(B\to K^{(\ast)}\nu\bar{\nu})_{\rm SM}},$
(74)

the NP contributions can be constrained. In the TVB model this ratio is
modified as

$\displaystyle R^{\nu\bar{\nu}}_{K^{(\ast)}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{ij}|\delta_{ij}C_{L}^{\rm SM}+\Delta
C_{L}^{ij}|^{2}}{3|C_{L}^{\rm SM}|^{2}},$ (75) $\displaystyle=$ $\displaystyle
1+\frac{2\sum_{i}C_{L}^{\rm SM}\Delta C_{L}^{ii}+\sum_{ij}|\Delta
C_{L}^{ij}|^{2}}{3|C_{L}^{\rm SM}|^{2}},$ (76)

From this expression, we can observe that diagonal leptonic couplings
$g^{\ell}_{\mu\mu}$ and $g^{\ell}_{\tau\tau}$ contribute to $b\to
s\nu_{\mu}\bar{\nu}_{\mu}$ (relevant for $b\to s\mu^{+}\mu^{-}$ data) and
$b\to s\nu_{\tau}\bar{\nu}_{\tau}$ (relevant for $b\to c\tau\bar{\nu}_{\tau}$
data), respectively. In addition, since the neutrino flavor is experimentally
unobservable in heavy meson experiments, it is also possible to induce the LFV
transitions $b\to s\nu_{\mu}\bar{\nu}_{\tau}$ (and
$\nu_{\tau}\bar{\nu}_{\mu}$) through the off-diagonal coupling
$g^{\ell}_{\mu\tau}$.

On the experimental side, the Belle experiment in 2017 obtained the following
ULs on the branching fractions ${\rm BR}(B\to K\nu\bar{\nu})<1.6\times
10^{-5}$ and ${\rm BR}(B\to K^{\ast}\nu\bar{\nu})<2.7\times 10^{-5}$
Grygier:2017tzo , resulting in limits on the ratios,
$R^{\nu\bar{\nu}}_{K}<3.9$ and $R^{\nu\bar{\nu}}_{K^{\ast}}<2.7$ ($90\%$
C.L.), respectively Grygier:2017tzo . In 2021, based on an inclusive tagging
technique, the Belle II experiment reported the bound ${\rm BR}(B^{+}\to
K^{+}\nu\bar{\nu})<4.1\times 10^{-5}$ at $90\%$ C.L. Belle-II:2021rof . A
combination of this new result with previous experimental results leads to the
weighted average ${\rm BR}(B^{+}\to K^{+}\nu\bar{\nu})=(1.1\pm 0.4)\times
10^{-5}$ Dattola:2021cmw . In turn, the ratio $R^{\nu\bar{\nu}}_{K^{+}}$ has
been calculated to be, $R^{\nu\bar{\nu}}_{K^{+}}=2.4\pm 0.9$ Browder:2021hbl .

The rare $B$ processes $B_{s}\to\tau^{+}\tau^{-}$ and $B\to K\tau^{+}\tau^{-}$
(induced via $b\to s\tau^{+}\tau^{-}$ transition) are expected to receive
significant NP impact. For the leptonic process $B_{s}\to\tau^{+}\tau^{-}$,
the SM branching ratio is shifted by the factor

${\rm BR}(B_{s}\to\tau^{+}\tau^{-})={\rm
BR}(B_{s}\to\tau^{+}\tau^{-})_{\text{SM}}\Bigg{|}1+\dfrac{\pi}{\sqrt{2}G_{F}\alpha_{\rm
em}V_{tb}V_{ts}^{\ast}C_{10}^{\rm
SM}}\dfrac{g^{q}_{sb}(g^{\ell}_{\tau\tau})^{\ast}}{M_{V}^{2}}\Bigg{|}^{2},$
(77)

where $C_{10}^{\rm SM}\simeq-4.3$. The strongest experimental bound on its
branching ratio has been obtained by the LHCb, ${\rm
BR}(B_{s}\to\tau^{+}\tau^{-})<6.8\times 10^{-3}$ at 95% confidence level
Aaij:2017xqt , while its SM predictions is ${\rm
BR}(B_{s}^{0}\to\tau^{+}\tau^{-})_{\rm SM}=(7.73\pm 0.49)\times 10^{-7}$
Bobeth:2013uxa . The bound is

$\displaystyle\dfrac{|g^{q}_{sb}(g^{\ell}_{\tau\tau})^{\ast}|}{M_{V}^{2}}$
$\displaystyle<$ $\displaystyle 0.56\ {\rm TeV}^{-2}.$ (78)

As concerns the semileptonic decay $B\to K\tau^{+}\tau^{-}$, an easy handle
numerical formula for the branching ratio (over the whole kinematic range for
the lepton pair invariant mass) has been obtained in Ref. Cornella:2019hct ,
for the case of a singlet vector leptoquark explanation of the $B$ meson
anomalies. Since the NP contribution is generated via the same operator, this
expression can be easily (but properly) translated to the TVB model, namely

${\rm BR}(B\to K\tau^{+}\tau^{-})\simeq 1.5\times 10^{-7}+1.4\times
10^{-3}\Big{(}\frac{1}{2\sqrt{2}G_{F}}\Big{)}\frac{{\rm
Re}[g^{q}_{sb}(g^{\ell}_{\tau\tau})^{\ast}]}{M_{V}^{2}}+3.5\Big{(}\frac{1}{2\sqrt{2}G_{F}}\Big{)}^{2}\frac{|g^{q}_{sb}(g^{\ell}_{\tau\tau})^{\ast}|^{2}}{M_{V}^{4}}.$
(79)

This decay channel has not been observed so far, and the present reported
bound is ${\rm BR}(B\to K\tau^{+}\tau^{-})<2.25\times 10^{-3}$ PDG2020 . We
obtained the following bound

$\frac{|g^{q}_{sb}(g^{\ell}_{\tau\tau})^{\ast}|}{M_{V}^{2}}<0.83\ {\rm
TeV}^{-2},$ (80)

that is weaker than the one get from $B_{s}\to\tau^{+}\tau^{-}$.

###  III.9 $\tau$ decays: $\tau\to 3\mu$,
$\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau}$, and $\tau\to\mu\phi$

It is known that the TVB model generates four-lepton operators
$(\bar{\mu}\gamma^{\alpha}P_{L}\tau)(\bar{\mu}\gamma_{\alpha}P_{L}\mu)$ and
$(\bar{\mu}\gamma^{\alpha}P_{L}\tau)(\bar{\nu}_{\tau}\gamma_{\alpha}P_{L}\nu_{\mu})$,
thus yielding to tree-level contributions to the leptonic $\tau$ decays,
$\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}\ (\tau\to 3\mu)$ and
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$, respectively
Bhattacharya:2016mcc ; Kumar:2018kmr . For the LFV decay $\tau\to 3\mu$, the
expression for the branching ratio can be written as

${\rm
BR}(\tau^{-}\to\mu^{-}\mu^{+}\mu^{-})=\frac{m_{\tau}^{5}}{1536\pi^{3}\Gamma_{\tau}}\frac{|g^{\ell}_{\mu\mu}g^{\ell}_{\mu\tau}|^{2}}{M_{V}^{4}},$
(81)

where $\Gamma_{\tau}$ is the total decay width of the $\tau$ lepton. The
current experimental UL obtained by Belle (at 90% CL) is ${\rm
BR}(\tau^{-}\to\mu^{-}\mu^{+}\mu^{-})<2.1\times 10^{-8}$ PDG2020 . This
corresponds to

$\frac{|g^{\ell}_{\mu\mu}g^{\ell}_{\mu\tau}|}{M_{V}^{2}}<1.13\times 10^{-2}\
{\rm TeV}^{-2}.$ (82)

The leptonic decay $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ is a lepton
flavor conserving and SM allowed process that receives tree-level contribution
from both $W^{\prime}$ (via lepton flavor conserving couplings) and
$Z^{\prime}$ (via LFV couplings) bosons Kumar:2018kmr . The branching ratio is
given by Kumar:2018kmr

${\rm BR}(\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau})={\rm
BR}(\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau})_{\rm
SM}\bigg{(}\bigg{|}1+\dfrac{1}{2\sqrt{2}G_{F}M_{V}^{2}}(2g^{\ell}_{\mu\mu}g^{\ell}_{\tau\tau}-|g^{\ell}_{\mu\tau}|^{2})\bigg{|}^{2}+\bigg{|}\dfrac{1}{2\sqrt{2}G_{F}M_{V}^{2}}|g^{\ell}_{\mu\tau}|^{2}\bigg{|}^{2}\bigg{)},$
(83)

where ${\rm BR}(\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau})_{\rm SM}=(17.29\pm
0.03)\%$ Pich:2013lsa . The $Z^{\prime}$ boson can also generates one-loop
corrections, which can be safely ignored. This value has to be compared with
the experimental value reported by PDG ${\rm
BR}(\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau})=(17.39\pm 0.04)\%$ PDG2020 .

Finally, the branching ratio of the LFV hadronic $\tau$ decay $\tau\to\mu\phi$
($\tau\to\mu s\bar{s}$ transition), can be expressed as Bhattacharya:2016mcc

${\rm
BR}(\tau^{-}\to\mu^{-}\phi)=\frac{f_{\phi}^{2}m_{\tau}^{3}}{128\pi\Gamma_{\tau}}\Big{(}1+2\frac{m_{\phi}^{2}}{m_{\tau}^{2}}\Big{)}\Big{(}1-\frac{m_{\phi}^{2}}{m_{\tau}^{2}}\Big{)}^{2}\frac{|g^{\ell}_{\mu\tau}g^{q}_{ss}|^{2}}{M_{V}^{4}},$
(84)

where $m_{\phi}$ and $f_{\phi}=(238\pm 3)$ MeV Kumar:2018kmr  are the $\phi$
meson mass and decay constant, respectively. Currently, the UL reported by
Belle on the branching ratio is ${\rm BR}(\tau^{-}\to\mu^{-}\phi)<2.3\times
10^{-8}$ Belle:2023ziz . The current UL produces the bound

$\frac{|g^{\ell}_{\mu\tau}g^{q}_{ss}|}{M_{V}^{2}}<9.4\times 10^{-3}\ {\rm
TeV}^{-3}.$ (85)

Since the $D^{0}-\bar{D}^{0}$ mixing imposes that $|g^{q}_{ss}|/M_{V}\leq
3.3\times 10^{-3}\ {\rm TeV^{-1}}$ (see Sec. III.5) the constraint from
$\tau\to\mu\phi$ can be easily fulfilled. We will not take into account this
LFV process in further TVB model analysis.

###  III.10 LHC bounds

LHC constraints are always important for models with non-zero $Z^{\prime}$
couplings to the SM particles Langacker:2008yv . In particular, in our study
it will set important constraints on the parametric space conformed by the TVB
couplings $(g^{q}_{bb},g^{\ell}_{\mu\mu})$ and
$(g^{q}_{bb},g^{\ell}_{\tau\tau})$. We consider the ATLAS search for high-mass
dilepton resonances in the mass range of 250 GeV to 6 TeV, in proton-proton
collisions at a center-of-mass energy of $\sqrt{s}=13$ TeV during Run 2 of the
LHC with an integrated luminosity of 139 fb-1 ATLAS:2019erb  (recently, the
CMS collaboration has also reported constraints for similar luminosities
CMS:2019tbu , basically identical to ATLAS ATLAS:2019erb ), and the data from
searches of $Z^{\prime}$ bosons decaying to tau pairs with an integrated
luminosity of 36.1 fb-1 from proton-proton collisions at $\sqrt{s}=13$ TeV
ATLAS:2017eiz . There are also searches for high-mass resonances in the
monolepton channels ($pp\to\ell\nu$) carried out by the ATLAS and CMS
ATLAS:2019lsy ; ATLASmonotau ; CMS:2022ncp . However, they provide weaker
bounds than those obtained from dilepton searches, and we will not take them
into account.

We obtain for benchmark mass value $M_{V}=1$ TeV the lower limit on the
parameter space from the intersection of the 95$\%$CL upper limit on the
cross-section from the ATLAS experiment ATLAS:2019erb ; ATLAS:2017eiz  with
the theoretical cross-section given in Ref. Erler:2011ud . Lower limits above
$4.5$ TeV apply to models with couplings to the first family, which it is not
our case. The strongest restrictions come from $Z^{\prime}$ production
processes in the $b\bar{b}$ annihilation and the subsequent $Z^{\prime}$ decay
into muons ($\mu^{+}\mu^{-}$) and taus ($\tau^{+}\tau^{-}$). Further details
are shown in Refs. Erler:2011ud ; Salazar:2015gxa ; Benavides:2018fzm . Let us
remark that within the TVB framework is also possible to consider the
annihilation between quarks with different flavors (namely, $g^{q}_{bs}$),
however, we anticipate that according to our phenomenological analysis in Sec.
IV this coupling is very small; therefore, we only consider production
processes without flavor changing neutral currents. In the next section we
will show that the TVB parameter space is limited by LHC constraints to
regions where the couplings of the leptons or the quarks are close to zero,
excluding the regions preferred by the $B$ meson anomalies and low-energy
flavor observables.

##  IV Analysis on the TVB parametric space

In this section we present the parametric space analysis of the TVB model
addressing a simultaneous explanation of the $b\to s\mu^{+}\mu^{-}$ and $b\to
c\tau\bar{\nu}_{\tau}$ data. We define the pull for the $i$-th observable as

${\rm pull}_{i}=\frac{\mathcal{O}^{\rm exp}_{i}-\mathcal{O}^{\rm
th}_{i}}{\Delta\mathcal{O}_{i}},$ (86)

where $\mathcal{O}^{\text{exp}}_{i}$ is the experimental measurement,
$\mathcal{O}^{\text{th}}_{i}\equiv\mathcal{O}^{\text{th}}_{i}(g^{q}_{bs},g^{q}_{bb},g^{\ell}_{\mu\mu},g^{\ell}_{\tau\tau},g^{\ell}_{\mu\tau})$
is the theoretical prediction that include the NP contributions, and
$\Delta\mathcal{O}_{i}=((\sigma^{\rm exp}_{i})^{2}+(\sigma^{\rm
th}_{i})^{2})^{1/2}$ corresponds to the combined experimental and theoretical
uncertainties. By means of the pull, we can compare the fitted values of each
observable to their measured values. The $\chi^{2}$ function is written as the
sum of squared pulls, i.e.,

$\chi^{2}=\sum_{i}^{N_{\rm obs}}({\rm pull}_{i})^{2},$ (87)

where the sum extends over the number of observables $(N_{\rm obs})$ to be
fitted. Our phenomenological analysis is based on the flavor observables
presented in the previous Sec. III. This all data set includes: $b\to
c\tau\bar{\nu}_{\tau}$ and $b\to s\mu^{+}\mu^{-}$ data, bottomonium ratios
$R_{\Upsilon(nS)}$, LFV decays ($B^{+}\to K^{+}\mu^{\pm}\tau^{\mp}$, $B^{0}\to
K^{\ast 0}\mu^{\pm}\tau^{\mp}$, $B_{s}\to\mu^{\pm}\tau^{\mp}$,
$\Upsilon(nS)\to\mu^{\pm}\tau^{\mp}$), rare $B$ decays ($B\to
K^{(\ast)}\nu\bar{\nu},B\to K\tau^{+}\tau^{-},B_{s}\to\tau^{+}\tau^{-}$),
$\tau$ decays ($\tau\to 3\mu$, $\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau}$), $\Delta
F=2$ processes, and neutrino trident production. We will study the impact of
the most recent LHCb measurements on the ratios $R(D^{(\ast)})$ LHCb2022 ;
LHCb:2023zxo ; LHCb2023 , allowing us to present an updated status of the TVB
model as an explanation to the $B$ meson anomalies. For such a purpose, we
will consider in our analysis the following three different sets of
observables,

  * •

All data with $R(D)_{\rm LHCb22}$ \+ $R(D^{\ast})_{\rm LHCb23}$,

  * •

All data with $R(D^{(\ast)})_{\rm LHCb22}$,

  * •

All data with $R(D^{(\ast)})_{\rm HFLAV23}$.

All these three sets contain a total number of observables $N_{\rm obs}=31$
and five free TVB parameters ($g^{q}_{bs}$, $g^{q}_{bb}$, $g^{\ell}_{\mu\mu}$,
$g^{\ell}_{\tau\tau}$, $g^{\ell}_{\mu\tau}$) to be fitted. The heavy TVB mass
will be fixed to the benchmark value $M_{V}=1\ {\rm TeV}$. Therefore, the
number of degrees of freedom is $N_{\rm dof}=26$.

Table 6: Best-fit point values and $1\sigma$ intervals of the five TVB couplings $(g^{q}_{bs},g^{q}_{bb},g^{\ell}_{\mu\mu},g^{\ell}_{\tau\tau},g^{\ell}_{\mu\tau})$ for the three different sets of observables and a benchmark mass value of $M_{V}=1\ {\rm TeV}$. TVB couplings | Best-fit point |  $1\sigma$ intervals
---|---|---
All data with $R(D)_{\rm LHCb22}$ \+ $R(D^{\ast})_{\rm LHCb23}$ :
$\chi^{2}_{\rm min}/N_{\rm dof}=0.63$, $p$-value $=93.7\%$
$g^{q}_{bs}$ | $-2.3\times 10^{-3}$ | $[-3.2,-1.6]\times 10^{-3}$
$g^{q}_{bb}$ | 0.73 | $[0.28,1.72]$
$g^{\ell}_{\mu\mu}$ | 0.20 | $[0.072,0.131]$
$g^{\ell}_{\tau\tau}$ | 0.49 | $[0.27,0.71]$
$g^{\ell}_{\mu\tau}$ | $\sim 0$ | $[-0.11,0.11]$
All data with $R(D^{(\ast)})_{\rm LHCb22}$ : $\chi^{2}_{\rm min}/N_{\rm
dof}=0.62$, $p$-value $=93.1\%$
$g^{q}_{bs}$ | $-3.2\times 10^{-3}$ | $[-4.4,-2.1]\times 10^{-3}$
$g^{q}_{bb}$ | 1.50 | $[0.74,2.24]$
$g^{\ell}_{\mu\mu}$ | 0.074 | $[0.052,0.095]$
$g^{\ell}_{\tau\tau}$ | 0.70 | $[0.45,0.94]$
$g^{\ell}_{\mu\tau}$ | $\sim 0$ | $[-0.15,0.15]$
All data with $R(D^{(\ast)})_{\rm HFLAV23}$ : $\chi^{2}_{\rm min}/N_{\rm
dof}=0.59$, $p$-value $=95.2\%$
$g^{q}_{bs}$ | $-3.2\times 10^{-3}$ | $[-4.4,-2.1]\times 10^{-3}$
$g^{q}_{bb}$ | 1.52 | $[1.09,1.94]$
$g^{\ell}_{\mu\mu}$ | 0.073 | $[0.052,0.095]$
$g^{\ell}_{\tau\tau}$ | 0.70 | $[0.53,0.88]$
$g^{\ell}_{\mu\tau}$ | $\sim 0$ | $[-0.14,0.14]$

For the three sets of observables we find the best-fit point values by
minimizing the $\chi^{2}$ function ($\chi^{2}_{\rm min}$). In Table 6 we
report our results of the best-fit point values and $1\sigma$ intervals of TVB
couplings. For each fit we also present in Table 6 the values of
$\chi^{2}_{\rm min}/N_{\rm dof}$ and its corresponding $p$-value to evaluate
the fit-quality. In general, it is found that the three sets of observables
provide an excellent fit of the data. In the quark sector, the TVB model
requires small $g^{q}_{bs}$ coupling, $|g^{q}_{bs}|\sim\mathcal{O}(10^{-3})$,
and opposite sign to $g^{\ell}_{\mu\mu}$ to be consistent with $b\to
s\mu^{+}\mu^{-}$ data ($C_{9}^{\mu\mu}=-C_{10}^{\mu\mu}$ solution) and
$B_{s}-\bar{B}_{s}$ mixing. On the other hand, large values for the bottom-
bottom coupling $g^{q}_{bb}\sim\mathcal{O}(1)$ are preferred. As for the
leptonic couplings, it is found that the lepton flavor conserving ones have a
similar size $g^{\ell}_{\mu\mu}\approx
g^{\ell}_{\tau\tau}\sim\mathcal{O}(10^{-1})$ for All data with $R(D)_{\rm
LHCb22}$ \+ $R(D^{\ast})_{\rm LHCb23}$, suggesting non-hierarchy pattern.
While for All data with $R(D^{(\ast)})_{\rm LHCb23}$ (with $R(D^{(\ast)})_{\rm
HFLAV23}$), these couplings exhibit a hierarchy
$g^{\ell}_{\tau\tau}>g^{\ell}_{\mu\mu}$. As LFV coupling concerns, the
obtained best-fit point values on $g^{\ell}_{\mu\tau}$ are negligible. Thus,
the TVB model do not lead to appreciable LFV effects. Last but no least, we
also probe higher mass values ($M_{V}>1\ {\rm TeV}$). We obtain that in order
to avoid large values on $g^{q}_{bb}$ coupling $(\sim\sqrt{4\pi})$, that would
put the perturbativity of the model into question, the TVB mass can be as
large as $M_{V}\sim 2$ TeV.

(a) All data with $R(D)_{\rm LHCb22}$ \+ $R(D^{\ast})_{\rm LHCb23}$

(b) All data with $R(D^{(\ast)})_{\rm LHCb23}$

(c) All data with $R(D^{(\ast)})_{\rm HFLAV23}$

Figure 1: $68\%$ (green) and $95\%$ (light-green) CL allowed regions for the
most relevant 2D parametric space of (a) All data with $R(D)_{\rm LHCb22}$ \+
$R(D^{\ast})_{\rm LHCb23}$, (b) All data with $R(D^{(\ast)})_{\rm LHCb22}$,
and (c) All data with $R(D^{(\ast)})_{\rm HFLAV23}$, respectively, for
$M_{V}=1\ {\rm TeV}$. In each plot we are marginalizing over the rest of the
parameters. The SM value is represented by the blue dot. The light-gray region
corresponds to LHC bounds at the $95\%$ CL. Perturbative region
($g^{q}_{bb}\geq\sqrt{4\pi})$) is represented by yellow color.

In Fig. 1, we show the allowed regions of the most relevant two-dimension (2D)
parametric space of (a) All data with $R(D)_{\rm LHCb22}$ \+ $R(D^{\ast})_{\rm
LHCb23}$, (b) All data with $R(D^{(\ast)})_{\rm LHCb22}$, and (c) All data
with $R(D^{(\ast)})_{\rm HFLAV23}$, respectively, for a benchmark TVB mass
$M_{V}=1\ {\rm TeV}$. The $68\%$ and $95\%$ CL regions are shown in green and
light-green colors, respectively. In each plot we are marginalizing over the
rest of the parameters. Furthermore, we include the LHC bounds (light-gray
regions) obtained from searches of high-mass dilepton (dimuon and ditau)
resonances at the ATLAS experiment ATLAS:2019erb ; ATLAS:2017eiz , as
discussed in Sec. III.10. For completeness, the perturbative region
($g^{q}_{bb}\geq\sqrt{4\pi})$) is represented by yellow color. It is observed
in the planes ($g^{q}_{bb},g^{\ell}_{\tau\tau}$) and
($g^{q}_{bb},g^{\ell}_{\mu\mu}$) for All data with $R(D^{(\ast)})_{\rm
HFLAV23}$ that the TVB model is seems to be strongly ruled out by the LHC
bounds. However, for All data with $R(D)_{\rm LHCb22}$ \+ $R(D^{\ast})_{\rm
LHCb23}$ (and with $R(D^{(\ast)})_{\rm LHCb22}$) that include the very recent
LHCb measurements LHCb2022 ; LHCb:2023zxo ; LHCb2023 , the TVB model can
provide a combined explanation of the $b\to c\tau\bar{\nu}_{\tau}$ and $b\to
s\mu^{+}\mu^{-}$ anomalies, in consistency with LHC bounds. Our analysis shows
that given the current experimental situation, particularly with LHCb, it is
premature to exclude the TVB model to addressing the $B$ meson anomalies.
Future improvements and new measurements on $b\to c\tau\bar{\nu}_{\tau}$ data
at the Belle II and LHCb experiments will be a matter of importance to test
the TVB model.

We close by mentioning that an analysis of the TVB model was previously
reported by Kumar, London, and Watanabe (KLW) by implementing the 2018 $b\to
c\tau\bar{\nu}_{\tau}$ and $b\to s\mu^{+}\mu^{-}$ data Kumar:2018kmr . KLW
found that the TVB model is excluded as a possible explanation of the $B$
meson anomalies due to the bound from LHC dimuon search (3.2 fb-1)
Kumar:2018kmr . Such a result is in agreement with ours for All data with
$R(D^{(\ast)})_{\rm HFLAV23}$ and considering recent LHC dimuon (139 fb-1) and
ditau (36.1 fb-1) searches. Unlike to KLW analysis, we have incorporated
several new observables and considered the recent available experimental
measurements and ULs. Thus, our present study extends, complements, and update
the previous analysis performed by KLW. We also extend the recent analysis
Garcia-Duque:2021qmg  where only the charged-current $b\to
c\tau\bar{\nu}_{\tau}$ anomaly was addressed within this framework.

###  IV.1 Implications to some flavor parametrizations

As a final step in our analysis, we will explore the implications to our
previous phenomenological analysis on TVB model to some flavor
parametrizations that have been already studied in the literature. For this we
consider scenarios in which the transformations involve only the second and
third generations Bhattacharya:2016mcc ; Calibbi:2015kma , as it was
previously discussed in Sec. II, we found that the equivalence in the quark
sector is Eq. (II.1), while for the leptonic sector we have Eq. (II.1). Taking
into account the $1\sigma$ range solutions of TVB couplings obtained in Table
6 (for the three sets of data), we get, in general, a large coupling
$g_{2}^{q}\sim\mathcal{O}(1)$ and a very small mixing angle $|\theta_{D}|\sim
10^{-3}$. Such a small mixing angle ($|\theta_{D}|\ll V_{cb}$) result is still
in agreement with previous analysis Bhattacharya:2016mcc ; Calibbi:2015kma .
On the contrary, in the leptonic sector, we obtained that because of $1\sigma$
range of the LFV coupling $g_{\mu\tau}^{\ell}$ it is not possible to find a
physical solution to the mixing angle $\theta_{L}$. As additional probe, we
have performed a global fit to the current $b\to s\mu^{+}\mu^{-}$ and $b\to
c\tau\bar{\nu}_{\tau}$ data, and the most relevant flavor observables, with
$(g_{2}^{q},g_{2}^{\ell},\theta_{D},\theta_{L})$ as free parameters. For a
fixed mass value $M_{V}=1\ {\rm TeV}$, we obtained a very poor fit
($\chi^{2}_{\rm min}/N_{\rm dof}\gg 1$), concluding that this kind of flavor
setup is not viable within the TVB model.

##  V Conclusions

We have presented an updated view of the TVB model as a simultaneous
explanation of the $B$ meson anomalies ($b\to c\tau\bar{\nu}_{\tau}$ and $b\to
s\mu^{+}\mu^{-}$ data). We performed a global fit of the TVB parameter space
with the most recent 2022 and 2023 data, including the LHCb measurements on
the charged-current LFU ratios $R(D^{(\ast)})$ and $R(\Lambda_{c})$. As
concerns $b\to s\mu^{+}\mu^{-}$ data, we taken into account the
$C^{bs\mu\mu}_{9}=-C^{bs\mu\mu}_{10}$ solution from global fit analysis
including the recent results on $R_{K^{(\ast)}}$ by LHCb and ${\rm
BR}(B_{s}\to\mu^{+}\mu^{-})$ by CMS. We have also included all relevant flavor
observables such as $B_{s}-\bar{B}_{s}$ mixing, neutrino trident production,
LFV decays ($B\to K^{(\ast)}\mu^{\pm}\tau^{\mp}$,
$B_{s}\to\mu^{\pm}\tau^{\mp}$, $\tau\to\mu\phi$,
$\Upsilon(nS)\to\mu^{\pm}\tau^{\mp}$), rare $B$ decays ($B\to
K^{(\ast)}\nu\bar{\nu},B\to K\tau^{+}\tau^{-},B_{s}\to\tau^{+}\tau^{-}$), and
bottomonium LFU ratios. We have confronted the allowed paramater space with
the LHC bounds from searches of high-mass dilepton resonances at the ATLAS
experiment.

Our analysis has shown that for a heavy TVB mass of 1 TeV and using all data
along with world averages values on $R(D^{(\ast)})$ reported by HFLAV, the TVB
model can accommodate the $b\to c\tau\bar{\nu}_{\tau}$ and $b\to
s\mu^{+}\mu^{-}$ anomalies (in consistency with other flavor observables), but
it seems to be strongly disfavoured by the LHC bounds. However, we obtained a
different situation when all data are combined with the very recent LHCb
measurements on $R(D^{(\ast)})$. The the $B$ meson anomalies can be addressed
within the TVB model in consistency with LHC constraints. We concluded that
new and improved $b\to c\tau\bar{\nu}_{\tau}$ data by LHCb and Belle II will
be required to really establish the viability of the TVB model.

We have also studied the consequences of our analysis of the TVB model to
flavor parametrizations in which the transformations involve only the second
and third generations. We obtained that such a flavor ansatz is not viable
within the TVB model.

###### Acknowledgements.

J. H. M. is grateful to Vicerrectoría de Investigación-Creación of Universidad
del Tolima for financial support of Project No. 290130517. E. R. acknowledges
financial support from the “Vicerrectoría de Investigaciones e Interacción
Social VIIS de la Universidad de Nariño,” Projects No. 1928 and No. 2172. We
are grateful to Hector Gisbert for his comments on LFV effects in the
dineutrino channels $B\to K^{(\ast)}\nu\bar{\nu}$.

## References

  * (1) D. London and J. Matias, $B$ Flavour Anomalies: 2021 Theoretical Status Report, Ann. Rev. Nucl. Part. Sci. 72, 37-68 (2022) [arXiv:2110.13270 [hep-ph]].
  * (2) J. Albrecht, D. van Dyk and C. Langenbruch, Flavour anomalies in heavy quark decays, Prog. Part. Nucl. Phys. 120, 103885 (2021) [arXiv:2107.04822 [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, 023001 (2019) [arXiv:1809.06229 [hep-ex]].
  * (4) R. Aaij et al. [LHCb], Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 113, 151601 (2014) [arXiv:1406.6482 [hep-ex]].
  * (5) R. Aaij et al. [LHCb], Test of lepton universality in beauty-quark decays, [arXiv:2103.11769 [hep-ex]].
  * (6) R. Aaij et al. [LHCb], Tests of lepton universality using $B^{0}\to K^{0}_{S}\ell^{+}\ell^{-}$ and $B^{+}\to K^{*+}\ell^{+}\ell^{-}$ decays, [arXiv:2110.09501 [hep-ex]].
  * (7) R. Aaij et al. [LHCb], Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 122, no.19, 191801 (2019) [arXiv:1903.09252 [hep-ex]].
  * (8) R. Aaij et al. [LHCb], Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays, JHEP 08, 055 (2017) [arXiv:1705.05802 [hep-ex]].
  * (9) [LHCb], Test of lepton universality in $b\rightarrow s\ell^{+}\ell^{-}$ decays, [arXiv:2212.09152 [hep-ex]].
  * (10) [LHCb], Measurement of lepton universality parameters in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ and $B^{0}\to K^{*0}\ell^{+}\ell^{-}$ decays, [arXiv:2212.09153 [hep-ex]].
  * (11) [CMS], Measurement of the B${}^{0}_{\mathrm{S}}$$\to$$\mu^{+}\mu^{-}$ decay properties and search for the B0$\to$$\mu^{+}\mu^{-}$ decay in proton-proton collisions at $\sqrt{s}$ = 13 TeV, [arXiv:2212.10311 [hep-ex]].
  * (12) R. Aaij et al. [LHCb], Measurement of Form-Factor-Independent Observables in the Decay $B^{0}\to K^{*0}\mu^{+}\mu^{-}$, Phys. Rev. Lett. 111, 191801 (2013) [arXiv:1308.1707 [hep-ex]].
  * (13) R. Aaij et al. [LHCb], Angular analysis of the $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity, JHEP 02, 104 (2016) [arXiv:1512.04442 [hep-ex]].
  * (14) R. Aaij et al. [LHCb], Measurement of $CP$-Averaged Observables in the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ Decay, Phys. Rev. Lett. 125, no.1, 011802 (2020) [arXiv:2003.04831 [hep-ex]].
  * (15) R. Aaij et al. [LHCb], Differential branching fraction and angular analysis of the decay $B_{s}^{0}\to\phi\mu^{+}\mu^{-}$, JHEP 07, 084 (2013) [arXiv:1305.2168 [hep-ex]].
  * (16) R. Aaij et al. [LHCb], Angular analysis and differential branching fraction of the decay $B^{0}_{s}\to\phi\mu^{+}\mu^{-}$, JHEP 09, 179 (2015) [arXiv:1506.08777 [hep-ex]].
  * (17) R. Aaij et al. [LHCb], Angular Analysis of the $B^{+}\rightarrow K^{\ast+}\mu^{+}\mu^{-}$ Decay, Phys. Rev. Lett. 126, no.16, 161802 (2021) [arXiv:2012.13241 [hep-ex]].
  * (18) J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl and D. M. Straub, $B$-decay discrepancies after Moriond 2019, Eur. Phys. J. C 80, no.3, 252 (2020) [arXiv:1903.10434 [hep-ph]].
  * (19) W. Altmannshofer and P. Stangl, New physics in rare B decays after Moriond 2021, Eur. Phys. J. C 81, no.10, 952 (2021) doi:10.1140/epjc/s10052-021-09725-1 [arXiv:2103.13370 [hep-ph]].
  * (20) M. Algueró, B. Capdevila, S. Descotes-Genon, J. Matias and M. Novoa-Brunet, $\bm{b\to s\ell\ell}$ global fits after Moriond 2021 results, [arXiv:2104.08921 [hep-ph]].
  * (21) M. Algueró, B. Capdevila, A. Crivellin, S. Descotes-Genon, P. Masjuan, J. Matias, M. Novoa Brunet and J. Virto, Emerging patterns of New Physics with and without Lepton Flavour Universal contributions, Eur. Phys. J. C 79, no.8, 714 (2019) [arXiv:1903.09578 [hep-ph]].
  * (22) L. S. Geng, B. Grinstein, S. Jäger, S. Y. Li, J. Martin Camalich and R. X. Shi, Implications of new evidence for lepton-universality violation in $b\to s\ell^{+}\ell^{-}$ decays, Phys. Rev. D 104, no.3, 035029 (2021) [arXiv:2103.12738 [hep-ph]].
  * (23) T. Hurth, F. Mahmoudi, D. M. Santos and S. Neshatpour, More Indications for Lepton Nonuniversality in $b\to s\ell^{+}\ell^{-}$, [arXiv:2104.10058 [hep-ph]].
  * (24) A. Angelescu, D. Bečirević, D. A. Faroughy, F. Jaffredo and O. Sumensari, Single leptoquark solutions to the B-physics anomalies, Phys. Rev. D 104, no.5, 055017 (2021) [arXiv:2103.12504 [hep-ph]].
  * (25) A. Carvunis, F. Dettori, S. Gangal, D. Guadagnoli and C. Normand, On the effective lifetime of $B_{s}\to\mu\mu\gamma$, JHEP 12, 078 (2021) [arXiv:2102.13390 [hep-ph]].
  * (26) A. Greljo, J. Salko, A. Smolkovič and P. Stangl, Rare $b$ decays meet high-mass Drell-Yan, [arXiv:2212.10497 [hep-ph]].
  * (27) M. Algueró, A. Biswas, B. Capdevila, S. Descotes-Genon, J. Matias and M. Novoa-Brunet, To (b)e or not to (b)e: No electrons at LHCb, [arXiv:2304.07330 [hep-ph]].
  * (28) J. P. Lees et al. [BaBar Collaboration], Evidence for an excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays, Phys. Rev. Lett. 109, 101802 (2012) [arXiv:1205.5442 [hep-ex]].
  * (29) J. P. Lees et al. [BaBar Collaboration], Measurement of an Excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ Decays and Implications for Charged Higgs Bosons, Phys. Rev. D 88, no. 7, 072012 (2013) [arXiv:1303.0571 [hep-ex]].
  * (30) M. Huschle et al. [Belle Collaboration], Measurement of the branching ratio of $\bar{B}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ relative to $\bar{B}\to D^{(\ast)}\ell^{-}\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle, Phys. Rev. D 92, no. 7, 072014 (2015) [arXiv:1507.03233 [hep-ex]].
  * (31) Y. Sato et al. [Belle Collaboration], Phys. Rev. D 94, no. 7, 072007 (2016) [arXiv:1607.07923 [hep-ex]].
  * (32) S. Hirose [Belle Collaboration], $\bar{B}\rightarrow D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ and Related Tauonic Topics at Belle, arXiv:1705.05100 [hep-ex].
  * (33) R. Aaij et al. [LHCb Collaboration], Measurement of the ratio of branching fractions $\mathcal{B}(\bar{B}^{0}\to D^{*+}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}^{0}\to D^{*+}\mu^{-}\bar{\nu}_{\mu})$, Phys. Rev. Lett. 115, no. 11, 111803 (2015) Erratum: [Phys. Rev. Lett. 115, no. 15, 159901 (2015)] [arXiv:1506.08614 [hep-ex]].
  * (34) R. Aaij et al. [LHCb Collaboration], Test of Lepton Flavor Universality by the measurement of the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ branching fraction using three-prong $\tau$ decays, Phys. Rev. D 97, no. 7, 072013 (2018) [arXiv:1711.02505 [hep-ex]].
  * (35) R. Aaij et al. [LHCb Collaboration], Measurement of the ratio of the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ and $B^{0}\to D^{*-}\mu^{+}\nu_{\mu}$ branching fractions using three-prong $\tau$-lepton decays, Phys. Rev. Lett. 120, no. 17, 171802 (2018) [arXiv:1708.08856 [hep-ex]].
  * (36) G. Caria et al. [Belle Collaboration], Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^{*})$ with a Semileptonic Tagging Method, Phys. Rev. Lett. 124 (2020) no.16, 161803 [arXiv:1910.05864 [hep-ex]].
  * (37) S. Hirose et al. [Belle Collaboration], Measurement of the $\tau$ lepton polarization and $R(D^{*})$ in the decay $\bar{B}\rightarrow D^{*}\tau^{-}\bar{\nu}_{\tau}$ with one-prong hadronic $\tau$ decays at Belle, Phys. Rev. D 97, no. 1, 012004 (2018) [arXiv:1709.00129 [hep-ex]].
  * (38) S. Hirose et al. [Belle Collaboration], Measurement of the $\tau$ lepton polarization and $R(D^{*})$ in the decay $\bar{B}\to D^{*}\tau^{-}\bar{\nu}_{\tau}$, Phys. Rev. Lett. 118, no. 21, 211801 (2017) [arXiv:1612.00529 [hep-ex]].
  * (39) R. Aaij et al. (LHCb Collaboration), 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, 121801 (2018) [arXiv:1711.05623 [hep-ex]].
  * (40) A. Abdesselam et al. [Belle Collaboration], Measurement of the $D^{\ast-}$ polarization in the decay $B^{0}\to D^{\ast-}\tau^{+}\nu_{\tau}$, arXiv:1903.03102 [hep-ex].
  * (41) Y. Amhis et al. [HFLAV], Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2021, [arXiv:2206.07501 [hep-ex]].
  * (42) LHCb Collaboration, First joint measurement of $R(D^{\ast})$ and $R(D^{0})$ at LHCb, https://indico.cern.ch/event/1187939/
  * (43) [LHCb], Measurement of the ratios of branching fractions $\mathcal{R}(D^{*})$ and $\mathcal{R}(D^{0})$, [arXiv:2302.02886 [hep-ex]].
  * (44) LHCb Collaboration, Measurement of $R(D^{\ast})$ with hadronic $\tau^{+}$ decays at $\sqrt{s}=$ 13 TeV by the LHCb collaboration, https://indico.cern.ch/event/1231797/.
  * (45) For updated results see HFLAV preliminary average of $R(D^{(\ast)})$ for Winter 2023 in https://hflav-eos.web.cern.ch/hflav-eos/semi/winter23_prel/html/RDsDsstar/RDRDs.html.
  * (46) J. Harrison et al. [LATTICE-HPQCD], $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, 222003 (2020) [arXiv:2007.06956 [hep-lat]].
  * (47) S. Iguro, T. Kitahara and R. Watanabe, Global fit to $b\to c\tau\nu$ anomalies 2022 mid-autumn, [arXiv:2210.10751 [hep-ph]].
  * (48) R. Alonso, B. Grinstein and J. Martin Camalich, Lifetime of $B_{c}^{-}$ Constrains Explanations for Anomalies in $B\to D^{(*)}\tau\nu$, Phys. Rev. Lett. 118, 081802 (2017). [arXiv:1611.06676 [hep-ph]]
  * (49) A. G. Akeroyd and C. H. Chen, Constraint on the branching ratio of $B_{c}\to\tau\nu$ from LEP1 and consequences for R(D(*)) anomaly, Phys. Rev. D 96, 075011 (2017). [arXiv:1708.04072 [hep-ph]].
  * (50) S. Kamali, New physics in inclusive semileptonic $B$ decays including nonperturbative corrections, Int. J. Mod. Phys. A 34, no.06n07, 1950036 (2019) [arXiv:1811.07393 [hep-ph]].
  * (51) R. Aaij et al. [LHCb], Observation of the decay $\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\tau^{-}\overline{\nu}_{\tau}$, Phys. Rev. Lett. 128, no.19, 191803 (2022) [arXiv:2201.03497 [hep-ex]].
  * (52) M. Fedele, M. Blanke, A. Crivellin, S. Iguro, T. Kitahara, U. Nierste and R. Watanabe, Impact of $\Lambda$b→$\Lambda$c$\tau$$\nu$ measurement on new physics in b→cl$\nu$ transitions, Phys. Rev. D 107, no.5, 055005 (2023) [arXiv:2211.14172 [hep-ph]].
  * (53) C. H. García-Duque, J. M. Cabarcas, J. H. Muñoz, N. Quintero and E. Rojas, Singlet vector leptoquark model facing recent LHCb and BABAR measurements,” Nucl. Phys. B 988, 116115 (2023) [arXiv:2209.04753 [hep-ph]].
  * (54) F. U. Bernlochner, Z. Ligeti, D. J. Robinson and W. L. Sutcliffe, Precise predictions for $\Lambda_{b}\to\Lambda_{c}$ semileptonic decays, Phys. Rev. D 99, no.5, 055008 (2019) [arXiv:1812.07593 [hep-ph]].
  * (55) L. Calibbi, A. Crivellin and T. Ota, Effective Field Theory Approach to $b\to s\ell\ell^{(\prime)}$, $B\to K^{(*)}\nu\overline{\nu}$ and $B\to D^{(*)}\tau\nu$ with Third Generation Couplings, Phys. Rev. Lett. 115, 181801 (2015) [arXiv:1506.02661 [hep-ph]].
  * (56) A. Greljo, G. Isidori and D. Marzocca, On the breaking of Lepton Flavor Universality in B decays, JHEP 07, 142 (2015) [arXiv:1506.01705 [hep-ph]].
  * (57) B. Bhattacharya, A. Datta, D. London and S. Shivashankara, Simultaneous Explanation of the $R_{K}$ and $R(D^{(*)})$ Puzzles, Phys. Lett. B 742, 370-374 (2015) [arXiv:1412.7164 [hep-ph]].
  * (58) D. A. Faroughy, A. Greljo and J. F. Kamenik, Confronting lepton flavor universality violation in B decays with high-$p_{T}$ tau lepton searches at LHC, Phys. Lett. B 764, 126 (2017). [arXiv:1609.07138 [hep-ph]]
  * (59) D. Buttazzo, A. Greljo, G. Isidori and D. Marzocca, B-physics anomalies: a guide to combined explanations, JHEP 11, 044 (2017) [arXiv:1706.07808 [hep-ph]].
  * (60) B. Bhattacharya, A. Datta, J. P. Guévin, D. London and R. Watanabe, Simultaneous explanation of the $R_{K}$ and $R_{D^{(*)}}$ puzzles: a model analysis, JHEP 01, 015 (2017) [arXiv:1609.09078 [hep-ph]].
  * (61) J. Kumar, D. London and R. Watanabe, 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, no.1, 015007 (2019) [arXiv:1806.07403 [hep-ph]].
  * (62) D. Guadagnoli, M. Reboud and O. Sumensari, A gauged horizontal $SU(2)$ symmetry and $R_{K^{(\ast)}}$, JHEP 11, 163 (2018) [arXiv:1807.03285 [hep-ph]].
  * (63) S. M. Boucenna, A. Celis, J. Fuentes-Martin, A. Vicente and J. Virto, Non-abelian gauge extensions for B-decay anomalies, Phys. Lett. B 760, 214-219 (2016) [arXiv:1604.03088 [hep-ph]].
  * (64) S. M. Boucenna, A. Celis, J. Fuentes-Martin, A. Vicente and J. Virto, Phenomenology of an $SU(2)\times SU(2)\times U(1)$ model with lepton-flavour non-universality, JHEP 12, 059 (2016) [arXiv:1608.01349 [hep-ph]].
  * (65) B. Capdevila, A. Crivellin, C. A. Manzari and M. Montull, Explaining $b\to s\ell^{+}\ell^{-}$ and the Cabibbo angle anomaly with a vector triplet, Phys. Rev. D 103, no.1, 015032 (2021) [arXiv:2005.13542 [hep-ph]].
  * (66) J. D. Gómez, N. Quintero and E. Rojas, Charged current $b\to c\tau\bar{\nu}_{\tau}$ anomalies in a general $W^{\prime}$ boson scenario, Phys. Rev. D 100, no.9, 093003 (2019) [arXiv:1907.08357 [hep-ph]].
  * (67) A. Datta, S. Kamali, S. Meinel and A. Rashed, Phenomenology of ${\Lambda}_{b}\to{\Lambda}_{c}\tau{\overline{\nu}}_{\tau}$ using lattice QCD calculations, JHEP 08, 131 (2017) [arXiv:1702.02243 [hep-ph]].
  * (68) E. Kou et al. [Belle-II], The Belle II Physics Book, PTEP 2019, no.12, 123C01 (2019) [erratum: PTEP 2020, no.2, 029201 (2020)] [arXiv:1808.10567 [hep-ex]]. Glattauer:2015teq
  * (69) R. Glattauer et al. [Belle], Measurement of the decay $B\to D\ell\nu_{\ell}$ in fully reconstructed events and determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$, Phys. Rev. D 93, no.3, 032006 (2016) [arXiv:1510.03657 [hep-ex]].
  * (70) A. Abdesselam et al. [Belle], Precise determination of the CKM matrix element $\left|V_{cb}\right|$ with $\bar{B}^{0}\to D^{*\,+}\,\ell^{-}\,\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle, [arXiv:1702.01521 [hep-ex]].
  * (71) D. Bečirević, F. Jaffredo, A. Peñuelas and O. Sumensari, New Physics effects in leptonic and semileptonic decays, JHEP 05, 175 (2021) [arXiv:2012.09872 [hep-ph]].
  * (72) C. Bobeth, M. Bordone, N. Gubernari, M. Jung and D. van Dyk, Lepton-flavour non-universality of ${\bar{B}}\rightarrow D^{*}\ell{{\bar{\nu}}}$ angular distributions in and beyond the Standard Model, Eur. Phys. J. C 81, no.11, 984 (2021) [arXiv:2104.02094 [hep-ph]].
  * (73) M. Bona et al. [UTfit], “New UTfit Analysis of the Unitarity Triangle in the Cabibbo-Kobayashi-Maskawa scheme,” [arXiv:2212.03894 [hep-ph]].
  * (74) R. L. Workman et al. [Particle Data Group], Review of Particle Physics, PTEP 2022, 083C01 (2022)
  * (75) M. T. Prim et al. [Belle], Search for $B^{+}\to\mu^{+}\,\nu_{\mu}$ and $B^{+}\to\mu^{+}\,N$ with inclusive tagging, Phys. Rev. D 101, no.3, 032007 (2020) [arXiv:1911.03186 [hep-ex]].
  * (76) C. H. García-Duque, J. H. Muñoz, N. Quintero and E. Rojas, Extra gauge bosons and lepton flavor universality violation in $\Upsilon$ and $B$ meson decays, Phys. Rev. D 103, no.7, 073003 (2021) [arXiv:2103.00344 [hep-ph]].
  * (77) D. Aloni, A. Efrati, Y. Grossman and Y. Nir, $\Upsilon$ and $\psi$ leptonic decays as probes of solutions to the $R_{D}^{(*)}$ puzzle, JHEP 06, 019 (2017) [arXiv:1702.07356 [hep-ph]].
  * (78) S. Descotes-Genon, S. Fajfer, J. F. Kamenik and M. Novoa-Brunet, Testing lepton flavor universality in $\Upsilon(4S)$ decays, Phys. Rev. D 103, no.11, 113009 (2021) [arXiv:2104.06842 [hep-ph]].
  * (79) P. del Amo Sanchez et al. [BaBar], Test of lepton universality in $\Upsilon(1S)$ decays at BaBar, Phys. Rev. Lett. 104, 191801 (2010) [arXiv:1002.4358 [hep-ex]].
  * (80) D. Besson et al. [CLEO], First Observation of $\Upsilon(3S)\to\tau^{+}\tau^{-}$ and Tests of Lepton Universality in Upsilon Decays, Phys. Rev. Lett. 98, 052002 (2007) [arXiv:hep-ex/0607019 [hep-ex]].
  * (81) J. P. Lees et al. [BaBar], Precision measurement of the ${\cal B}(\Upsilon(3S)\to\tau^{+}\tau^{-})/{\cal B}(\Upsilon(3S)\to\mu^{+}\mu^{-})$ ratio, Phys. Rev. Lett. 125, 241801 (2020) [arXiv:2005.01230 [hep-ex]].
  * (82) S. Patra et al. [Belle], Search for charged lepton flavor violating decays of $\Upsilon(1S)$, JHEP 05, 095 (2022) [arXiv:2201.09620 [hep-ex]].
  * (83) L. Di Luzio, M. Kirk, A. Lenz and T. Rauh, $\Delta M_{s}$ theory precision confronts flavour anomalies, JHEP 12, 009 (2019) [arXiv:1909.11087 [hep-ph]].
  * (84) L. Di Luzio, M. Kirk and A. Lenz, Updated $B_{s}$-mixing constraints on new physics models for $b\to s\ell^{+}\ell^{-}$ anomalies, Phys. Rev. D 97, no.9, 095035 (2018) [arXiv:1712.06572 [hep-ph]].
  * (85) A. K. Alok, N. R. S. Chundawat and D. Kumar, Impact of $b\rightarrow s\ell\ell$ anomalies on rare charm decays in non-universal $Z^{\prime}$ models, Eur. Phys. J. C 82, no.1, 30 (2022) [arXiv:2110.12451 [hep-ph]].
  * (86) W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams, Phys. Rev. Lett. 113, 091801 (2014) [arXiv:1406.2332 [hep-ph]].
  * (87) R. Aaij et al. [LHCb], Search for the lepton flavour violating decay $B^{+}\rightarrow K^{+}\mu^{-}\tau^{+}$ using $B_{s2}^{*0}$ decays, JHEP 06, 129 (2020) [arXiv:2003.04352 [hep-ex]].
  * (88) [LHCb], Search for the lepton-flavour violating decays $B^{0}\to K^{*0}\tau^{\pm}\mu^{\mp}$, [arXiv:2209.09846 [hep-ex]].
  * (89) W. G. Parrott et al. [HPQCD], Standard Model predictions for B→K$\ell$+$\ell$-, B→K$\ell$1-$\ell$2+ and B→K$\nu$$\nu$¯ using form factors from Nf=2+1+1 lattice QCD, Phys. Rev. D 107, no.1, 014511 (2023) [arXiv:2207.13371 [hep-ph]].
  * (90) R. Aaij et al. [LHCb], Search for the lepton-flavour-violating decays $B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ and $B^{0}\to\tau^{\pm}\mu^{\mp}$, Phys. Rev. Lett. 123, no.21, 211801 (2019) [arXiv:1905.06614 [hep-ex]].
  * (91) R. Bause, H. Gisbert, M. Golz and G. Hiller, Lepton universality and lepton flavor conservation tests with dineutrino modes, Eur. Phys. J. C 82, no.2, 164 (2022) [arXiv:2007.05001 [hep-ph]].
  * (92) R. Bause, H. Gisbert, M. Golz and G. Hiller, Interplay of dineutrino modes with semileptonic rare B-decays, JHEP 12, 061 (2021) [arXiv:2109.01675 [hep-ph]].
  * (93) T. E. Browder, N. G. Deshpande, R. Mandal and R. Sinha, Impact of $B\to K\nu\bar{\nu}$ measurements on beyond the Standard Model theories, Phys. Rev. D 104, no.5, 053007 (2021) [arXiv:2107.01080 [hep-ph]].
  * (94) X. G. He and G. Valencia, $R^{\nu}_{K^{(\ast)}}$ and non-standard neutrino interactions, Phys. Lett. B 821, 136607 (2021) [arXiv:2108.05033 [hep-ph]].
  * (95) A. J. Buras, J. Girrbach-Noe, C. Niehoff and D. M. Straub, $B\to{K}^{\left(\ast\right)}\nu\overline{\nu}$ decays in the Standard Model and beyond, JHEP 02, 184 (2015) [arXiv:1409.4557 [hep-ph]].
  * (96) J. Grygier et al. [Belle], Search for $\bm{B\to h\nu\bar{\nu}}$ decays with semileptonic tagging at Belle, Phys. Rev. D 96, no.9, 091101 (2017) [arXiv:1702.03224 [hep-ex]].
  * (97) F. Abudinén et al. [Belle-II], Search for $B^{+}\to K^{+}\nu\bar{\nu}$ Decays Using an Inclusive Tagging Method at Belle II, Phys. Rev. Lett. 127, no.18, 181802 (2021) [arXiv:2104.12624 [hep-ex]].
  * (98) F. Dattola [Belle-II], Search for $B^{+}\to K^{+}\nu\bar{\nu}$ decays with an inclusive tagging method at the Belle II experiment, [arXiv:2105.05754 [hep-ex]].
  * (99) R. Aaij et al. [LHCb], Search for the decays $B_{s}^{0}\to\tau^{+}\tau^{-}$ and $B^{0}\to\tau^{+}\tau^{-}$, Phys. Rev. Lett. 118, no.25, 251802 (2017) [arXiv:1703.02508 [hep-ex]].
  * (100) C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou and M. Steinhauser, $B_{s,d}\to l^{+}l^{-}$ in the Standard Model with Reduced Theoretical Uncertainty, Phys. Rev. Lett. 112, 101801 (2014) [arXiv:1311.0903 [hep-ph]].
  * (101) J. Albrecht, F. Bernlochner, M. Kenzie, S. Reichert, D. Straub and A. Tully, Future prospects for exploring present day anomalies in flavour physics measurements with Belle II and LHCb, [arXiv:1709.10308 [hep-ph]].
  * (102) C. Cornella, J. Fuentes-Martin and G. Isidori, Revisiting the vector leptoquark explanation of the B-physics anomalies, JHEP 07, 168 (2019) [arXiv:1903.11517 [hep-ph]].
  * (103) A. Pich, Precision Tau Physics, Prog. Part. Nucl. Phys. 75, 41-85 (2014) [arXiv:1310.7922 [hep-ph]].
  * (104) N. Tsuzuki et al. [Belle], Search for lepton-flavor-violating $\tau$ decays into a lepton and a vector meson using the full Belle data sample, [arXiv:2301.03768 [hep-ex]].
  * (105) P. Langacker, The Physics of Heavy $Z^{\prime}$ Gauge Bosons, Rev. Mod. Phys. 81, 1199-1228 (2009) [arXiv:0801.1345 [hep-ph]].
  * (106) G. Aad et al. [ATLAS], Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector, Phys. Lett. B 796, 68-87 (2019) [arXiv:1903.06248 [hep-ex]].
  * (107) [CMS], Search for a narrow resonance in high-mass dilepton final states in proton-proton collisions using 140$~{}\mathrm{fb}^{-1}$ of data at $\sqrt{s}=13~{}\mathrm{TeV}$, CMS-PAS-EXO-19-019.
  * (108) M. Aaboud et al. [ATLAS], Search for additional heavy neutral Higgs and gauge bosons in the ditau final state produced in 36 fb-1 of pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, JHEP 01 (2018), 055 [arXiv:1709.07242 [hep-ex]].
  * (109) G. Aad et al. [ATLAS], Search for a heavy charged boson in events with a charged lepton and missing transverse momentum from $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, Phys. Rev. D 100, no.5, 052013 (2019) [arXiv:1906.05609 [hep-ex]].
  * (110) G. Aad et al. [ATLAS], Search for high-mass resonances in final states with a tau lepton and missing transverse momentum with the ATLAS detector, ATLAS-CONF-2021-025.
  * (111) [CMS], Search for new physics in the $\tau$ lepton plus missing transverse momentum final state in proton-proton collisions at $\sqrt{s}$ = 13 TeV, [arXiv:2212.12604 [hep-ex]].
  * (112) J. Erler, P. Langacker, S. Munir and E. Rojas, $Z^{\prime}$ Bosons at Colliders: a Bayesian Viewpoint, JHEP 11, 076 (2011) [arXiv:1103.2659 [hep-ph]].
  * (113) C. Salazar, R. H. Benavides, W. A. Ponce and E. Rojas, LHC Constraints on 3-3-1 Models, JHEP 07, 096 (2015) [arXiv:1503.03519 [hep-ph]].
  * (114) R. H. Benavides, L. Muñoz, W. A. Ponce, O. Rodríguez and E. Rojas, Electroweak couplings and LHC constraints on alternative Z’ models in $E_{6}$, Int. J. Mod. Phys. A 33, no.35, 1850206 (2018) [arXiv:1801.10595 [hep-ph]].