††thanks: (Corresponding author)

# Extra gauge bosons and lepton flavor universality violation in $\Upsilon$
and $B$ meson decays

Cristian H. García-Duque<EMAIL_ADDRESS>Programa de Física,
Universidad del Quindío, Carrera 15 Calle 12 Norte, Código Postal 630004,
Armenia, 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  Departamento de Física, Universidad
del Tolima, Código Postal 730006299, Ibagué, Colombia  Eduardo Rojas
<EMAIL_ADDRESS>Departamento de Física, Universidad de Nariño, A.A. 1175,
San Juan de Pasto, Colombia

###### Abstract

Lepton flavor universality can be tested through the ratio of semileptonic $B$
meson decays and leptonic $\Upsilon$ meson decays, with
$\Upsilon\equiv\Upsilon(nS)$ ($n=1,2,3$). For the charged-current transitions
$b\to c\tau\bar{\nu}_{\tau}$, discrepancies between the experiment and the
Standard Model (SM) have been observed in recent years by different flavor
facilities such as BABAR, Belle, and LHCb. While for the neutral-current
transitions $b\bar{b}\to\tau\bar{\tau}$, the BABAR experiment reported
recently a new measurement of leptonic decay ratio $R_{\Upsilon(3S)}={\rm
BR}(\Upsilon(3S)\to\tau^{+}\tau^{-})/{\rm BR}(\Upsilon(3S)\to\mu^{+}\mu^{-})$,
showing an agreement with the SM at the $1.8\sigma$ level. In light of this
new BABAR result and regarding the connection between new physics (NP)
interpretations to the charged-current $b\to c\tau\bar{\nu}_{\tau}$ anomalies
and neutral-current $b\bar{b}\to\tau\bar{\tau}$ processes, in this study, we
revisit the NP consequences of this measurement within a simplified model with
extra massive gauge bosons that coupled predominantly to left-handed leptons
of the third-generation. We show that the BABAR measurement of
$R_{\Upsilon(3S)}$ cannot easily be accommodated (within its experimental
$1\sigma$ range) together with the other $b\to c\tau\bar{\nu}_{\tau}$ data,
hinting toward a new anomalous observable.

##  I Introduction

In recent years, tantalizing hints of lepton flavor universality (LFU)
violation have been suggested by the experiments BABAR, Belle, and LHCb in the
measurements of the ratio of semileptonic $B$ meson decays Lees:2012xj ;
Lees:2013uzd ; Huschle:2015rga ; Sato:2016svk ; Hirose:2017vbz ; Aaij:2015yra
; Aaij:2017deq ; Aaij:2017uff ; Belle:2019rba ; Hirose:2017dxl ;
Hirose:2016wfn ; Amhis:2019ckw ; HFLAVsummer ; Aaij:2017tyk

$\displaystyle R(D^{(\ast)})$ $\displaystyle=$ $\displaystyle\dfrac{{\rm
BR}(B\to D^{(\ast)}\tau\bar{\nu}_{\tau})}{{\rm BR}(B\to
D^{(\ast)}\ell^{\prime}\bar{\nu}_{\ell^{\prime}})}\ \ \ (\ell^{\prime}=e\ {\rm
or}\ \mu),$ (1) $\displaystyle R(J/\psi)$ $\displaystyle=$
$\displaystyle\dfrac{{\rm BR}(B_{c}\to J/\psi\tau\bar{\nu}_{\tau})}{{\rm
BR}(B_{c}\to J/\psi\mu\bar{\nu}_{\mu})}.$ (2)

The latest 2019 world averages values reported by the Heavy Flavor Averaging
Group (HFLAV) on the measurements of $R(D^{(\ast)})$ Amhis:2019ckw ;
HFLAVsummer  and the LHCb results on $R(J/\psi)$ Aaij:2017tyk ;
Harrison:2020nrv ; Harrison:2020gvo ,

$\displaystyle R(D)$ $\displaystyle=$
$\displaystyle\begin{cases}\text{HFLAV:}\ 0.340\pm 0.027\pm
0.013\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Amhis:2019ckw,HFLAVsummer}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.299\pm 0.003\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Amhis:2019ckw,HFLAVsummer}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(3) $\displaystyle R(D^{\ast})$ $\displaystyle=$
$\displaystyle\begin{cases}\text{HFLAV:}\ 0.295\pm 0.011\pm
0.008\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Amhis:2019ckw,HFLAVsummer}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.258\pm 0.005\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Amhis:2019ckw,HFLAVsummer}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(4) $\displaystyle R(J/\psi)$ $\displaystyle=$
$\displaystyle\begin{cases}\text{LHCb:}\ 0.71\pm 0.17\pm
0.18\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aaij:2017tyk}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.283\pm 0.048\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Harrison:2020nrv,Harrison:2020gvo}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(5)

exhibit a deviation with respect to the Standard Model (SM) expectations by
$1.4\sigma$, $2.5\sigma$, and $1.8\sigma$, respectively. SM predictions for
$R(D^{(\ast)})$ are taken from the average values obtained by HFLAV
HFLAVsummer , while for $R(J/\psi)$ we consider the recent lattice QCD
calculations Harrison:2020nrv ; Harrison:2020gvo . Furthermore, polarization
observables such as the $\tau$ lepton polarization $P_{\tau}(D^{\ast})$ and
the longitudinal polarization of the $D^{*}$ meson $F_{L}(D^{\ast})$ related
with the channel $\bar{B}\to D^{\ast}\tau\bar{\nu}_{\tau}$ have been observed
in the Belle experiment Hirose:2017dxl ; Hirose:2016wfn ; Abdesselam:2019wbt ,

$\displaystyle P_{\tau}(D^{\ast})$ $\displaystyle=$
$\displaystyle\begin{cases}\text{Belle:}\ -0.38\pm
0.51{}^{+0.21}_{-0.16}\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hirose:2017dxl,Hirose:2016wfn}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ -0.497\pm 0.013\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Tanaka:2012nw}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(6) $\displaystyle F_{L}(D^{\ast})$ $\displaystyle=$
$\displaystyle\begin{cases}\text{Belle:}\ 0.60\pm 0.08\pm
0.035\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Abdesselam:2019wbt}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.46\pm 0.04\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Alok:2016qyh}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(7)

and also present a disagreement respect with the corresponding SM predictions
Tanaka:2012nw ; Alok:2016qyh . Additionally, strong constraints from the upper
limits on the branching ratio of the tauonic $B_{c}$ decay, ${\rm
BR}(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})\lesssim 30\%$ and $10\%$, imposed by
the lifetime of $B_{c}$ meson Alonso:2016oyd  and the LEP data taken at the
$Z$ peak Akeroyd:2017mhr , respectively, have to be taken into account.

All these measurements on $b\to c\tau\bar{\nu}_{\tau}$ data point toward LFU
violation and they are generally referred to as $b\to c\tau\bar{\nu}_{\tau}$
anomalies. Several model-independent studies of the effect of new physics (NP)
operators regarding the most general dimension-six effective Lagrangian with
the most recent $b\rightarrow c\tau\bar{\nu}_{\tau}$ data have been explored
Iguro:2020keo ; Iguro:2020cpg ; Asadi:2019xrc ; Murgui:2019czp ;
Mandal:2020htr ; Cheung:2020sbq ; Sahoo:2019hbu ; Shi:2019gxi ;
Bardhan:2019ljo ; Blanke:2018yud ; Blanke:2019qrx ; Alok:2019uqc ;
Huang:2018nnq ; Jung:2018lfu ; Kumbhakar:2020jdz ; Angelescu:2018tyl ;
Feruglio:2018fxo ; Iguro:2018vqb . Sharing the same Lorentz structure as the
SM, NP arising from left-handed vector operator
$(\bar{c}\gamma_{\mu}P_{L}b)(\bar{\tau}\gamma^{\mu}P_{L}\nu_{\tau})$ is still
a preferred and feasible solution to address the anomalies, providing a good
fit to the data Iguro:2020keo ; Iguro:2020cpg ; Asadi:2019xrc ; Murgui:2019czp
; Mandal:2020htr ; Cheung:2020sbq ; Sahoo:2019hbu ; Shi:2019gxi ;
Bardhan:2019ljo ; Blanke:2018yud ; Blanke:2019qrx ; Alok:2019uqc ;
Huang:2018nnq ; Jung:2018lfu ; Kumbhakar:2020jdz ; Angelescu:2018tyl ;
Feruglio:2018fxo ; Iguro:2018vqb . Different NP scenarios can be generated via
this semi-tauonic operator. One interesting possibility to accommodate the
anomalies consists in considering an extra left-handed gauge boson
$W^{\prime}$ Bhattacharya:2014wla ; Greljo:2015mma ; Faroughy:2016osc ;
Abdullah:2018ets ; Greljo:2018tzh ; Dasgupta:2018nzt ; Boucenna:2016qad ;
Boucenna:2016wpr ; Buttazzo:2017ixm ; Bhattacharya:2016mcc ; Kumar:2018kmr ;
Gomez:2019xfw ; Iguro:2018fni . The opening works suggesting in the literature
a $SU(2)_{L}$ triplet of massive vectors mostly coupled to the left-handed
fermions of the third-generation (referred to as vector triplet model or
vector boson model) were presented in Refs. Bhattacharya:2014wla ;
Greljo:2015mma ; Faroughy:2016osc 111Let us notice that a simultaneous
explanation of both the $b\to c\tau\bar{\nu}_{\tau}$ and $b\to
s\mu^{+}\mu^{-}$ anomalies have been also discussed within the vector boson
model in Refs. Bhattacharya:2014wla ; Greljo:2015mma ; Bhattacharya:2016mcc ;
Kumar:2018kmr ; however, this approach is beyond the scope of the present
work.. It was found that although the model can accommodate the
$R(D^{(\ast)})$ anomalies, the framework is severely constrained by the direct
searches of neutral resonances decaying into $\tau^{+}\tau^{-}$ pairs at ATLAS
and CMS Greljo:2015mma ; Faroughy:2016osc . Different $W^{\prime}$ boson
scenarios (either UV completions and simplified models) have also been studied
Abdullah:2018ets ; Greljo:2018tzh ; Dasgupta:2018nzt ; Boucenna:2016qad ;
Boucenna:2016wpr ; Buttazzo:2017ixm ; Bhattacharya:2016mcc ; Kumar:2018kmr ;
Gomez:2019xfw ; Iguro:2018fni  and complementary tests of these models with
the searches for heavy $\tau\nu$ resonances performed at the LHC, showed an
agreement with the constraints from ATLAS and CMS data (see, for instance,
Refs. Iguro:2018fni ; Abdullah:2018ets ; Greljo:2018tzh ; Marzocca:2020ueu ).

Additionally, alternative approaches regarding $W^{\prime}$ bosons associated
with pure right-handed currents (involving a right-handed neutrino) have been
discussed recently in the literature within different NP realizations
Greljo:2018tzh ; Gomez:2019xfw ; Iguro:2018fni ; Hayreter:2019dzc ; He:2012zp
; He:2017bft ; Carena:2018cow ; Babu:2018vrl ; Asadi:2018wea ; Greljo:2018ogz
; Robinson:2018gza ; Asadi:2018sym ; Azatov:2018kzb ; Cvetic:2017gkt .
Nevertheless, some recent analyses have shown that this right-handed neutrino
interpretation seems to be disfavored by the LHC data Shi:2019gxi ;
Greljo:2018tzh .

On the other hand, LFU can also be tested through the ratio of leptonic decays
of bottomonium meson $\Upsilon(nS)$ Aloni:2017eny

$R_{\Upsilon(nS)}\equiv\frac{{\rm BR}(\Upsilon(nS)\to\tau^{+}\tau^{-})}{{\rm
BR}(\Upsilon(nS)\to\ell^{+}\ell^{-})},$ (8)

with $n=1,2,3$ and $\ell=\mu,e$, providing a clean theoretical environment.
Experimentally, the BABAR and CLEO Collaborations have reported the values
delAmoSanchez:2010bt ; Besson:2006gj

$\displaystyle R_{\Upsilon(1S)}$ $\displaystyle=$
$\displaystyle\begin{cases}\text{BABAR-10:}\ 1.005\pm 0.013\pm
0.022\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{delAmoSanchez:2010bt}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.9924\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aloni:2017eny}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(9) $\displaystyle R_{\Upsilon(2S)}$ $\displaystyle=$
$\displaystyle\begin{cases}\text{CLEO-07:}\ 1.04\pm 0.04\pm
0.05\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Besson:2006gj}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.9940\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aloni:2017eny}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(10) $\displaystyle R_{\Upsilon(3S)}$ $\displaystyle=$
$\displaystyle\begin{cases}\text{CLEO-07:}\ 1.05\pm 0.08\pm
0.05\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Besson:2006gj}{\@@citephrase{(}}{\@@citephrase{)}}}},\\\
\text{SM:}\ 0.9948\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aloni:2017eny}{\@@citephrase{(}}{\@@citephrase{)}}}},\end{cases}$
(11)

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 by $0.5\sigma,0.8\sigma$, and $0.6\sigma$,
respectively. Recently, in 2020 the BABAR experiment has released a new
measurement on the ratio $R_{\Upsilon(3S)}$ Lees:2020kom , whose value is

$R_{\Upsilon(3S)}^{\rm BABAR-20}=0.966\pm 0.008\pm 0.014,$ (12)

which improves the precision of the experimental value previously obtained by
CLEO Besson:2006gj . Despite this improvement, the new value is below the SM
expectation and shows an agreement at the $1.8\sigma$ level Lees:2020kom , in
higher tension than CLEO. Moreover, averaging the CLEO-07 Besson:2006gj  and
BABAR-20 Lees:2020kom  measurements we obtain

$R_{\Upsilon(3S)}^{\rm Ave}=0.968\pm 0.016,$ (13)

which deviates at the $1.7\sigma$ level with respect to the SM prediction
(uncertainties were taken in quadrature). Motivated by the tension generated
by the new BABAR measurement on $R_{\Upsilon(3S)}$, it is intriguing to study
its possible NP implications. As additional motivation, it is known that new
physics scenarios (with left-handed neutrinos) aiming to provide an
explanation to the $R(D^{(\ast)})$ anomalies also induce inevitable effects in
the leptonic decay ratio $R_{\Upsilon(nS)}$ Aloni:2017eny . The connection
between charged-current $b\to c\tau\bar{\nu}_{\tau}$ and neutral-current
$b\bar{b}\to\tau\bar{\tau}$ processes was first pointed out by the authors of
Ref. Faroughy:2016osc , in which they performed a recast of existing
$\tau^{+}\tau^{-}$ resonance searches at the CMS and ATLAS experiments,
allowing one to set constraints on different simplified models addressing the
$R(D^{(\ast)})$ anomalies.

Keeping in mind the correlation between NP solutions to the charged-current
$b\to c\tau\bar{\nu}_{\tau}$ anomalies and neutral-current
$b\bar{b}\to\tau\bar{\tau}$ processes Faroughy:2016osc ; Aloni:2017eny , and
to the light of the very recent BABAR result on $R_{\Upsilon(3S)}$
Lees:2020kom , in this work we present a reanalysis of the extra gauge bosons
within the vector triplet model that preferentially couples to third-
generation fermions Greljo:2015mma ; Faroughy:2016osc . A previous analysis
addressing the $R(D^{(\ast)})$ anomalies and the complementary
$R_{\Upsilon(nS)}$ in this model was presented in Ref. Aloni:2017eny , in
which the authors found within $95\%$ confidence level the numerical values
for the Wilson coefficients that minimize the observed anomaly in
$R(D^{(\ast)})$, and the corresponding predictions for $R_{\Upsilon(nS)}$.
This study was implemented by considering the 2016 HFLAV averages
Amhis:2016xyh , which differ from the most recent 2019 HFLAV ones
Amhis:2019ckw ; HFLAVsummer . Here, by means of a different approach we carry
out a robust phenomenological analysis of the parametric space of gauge
couplings allowed by charged-current $b\to c\tau\bar{\nu}_{\tau}$ and
$R_{\Upsilon(nS)}$ data. Particularly, for the $b\to c\tau\bar{\nu}_{\tau}$
data, we include the polarizations of $D^{*}$ and the tau lepton associated
with $\bar{B}\to D^{\ast}\tau\bar{\nu}_{\tau}$, the ratio $R(J/\psi)$, and the
upper limit on BR$(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})$, and we incorporate
the forthcoming sensitivity of Belle II on $R(D^{(\ast)})$ measurements. In
that sense, our work complements and extends the previous analysis performed
in Aloni:2017eny . We will show that the vector triplet model is in conflict
with the BABAR measurement of $R_{\Upsilon(3S)}$ and the $1\sigma$ range
uncertainties cannot be explained in simultaneity with $b\to
c\tau\bar{\nu}_{\tau}$ data.

The outline of this paper is organized as follows. In Sec. II, we briefly
present the main features of the left-handed vector bosons model. A
phenomenological analysis of the parametric space of gauge couplings allowed
by charged-current and neutral-current data is presented in Sec. III. The main
concluding remarks of this work are given in Sec. IV.

##  II Left-handed vector bosons model

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 third-generation Greljo:2015mma ;
Faroughy:2016osc . The Lagrangian describing the interactions between fermions
and vector boson is Greljo:2015mma ; Faroughy:2016osc

$\mathcal{L}^{\rm LH-
VB}=g_{b}\bar{Q}_{3}\frac{\sigma_{a}}{2}\gamma^{\mu}W^{a}_{\mu}Q_{3}+g_{\tau}\bar{L}_{3}\frac{\sigma_{a}}{2}\gamma^{\mu}W^{a}_{\mu}L_{3},$
(14)

where $Q_{3}=(V_{cb}c_{L},\ b_{L})^{T}$ and $L_{3}=(\nu_{\tau L},\
\tau_{L})^{T}$ are the LH quark and lepton doublets, $\sigma_{a}$ ($a=1,2,3$)
are the Pauli matrices, $V_{cb}$ is the associated Cabbibo-Kobayashi-Maskawa
(CKM) matrix element, and $g_{b}$ and $g_{\tau}$ are the corresponding
couplings of LH quarks and leptons to vector bosons, respectively. The down-
type quark and charged-lepton mass eigenstate basis have been adopted for the
LH fermion multiplets. After the heavy vector bosons are integrating out, the
relevant charged-current $b\to c\tau\bar{\nu}_{\tau}$ and neutral-current
$b\bar{b}\to\tau\bar{\tau}$ operators are given by Greljo:2015mma

$\displaystyle\mathcal{L}_{\rm CC}$ $\displaystyle=$
$\displaystyle-\frac{g_{b}g_{\tau}}{2M^{2}_{W^{\prime}}}V_{cb}(\bar{c}\gamma_{\mu}P_{L}b)(\bar{\tau}\gamma^{\mu}P_{L}\nu_{\tau})+\
{\rm H.c.},$ (15) $\displaystyle\mathcal{L}_{\rm NC}$ $\displaystyle=$
$\displaystyle-\frac{g_{b}g_{\tau}}{4M^{2}_{Z^{\prime}}}(\bar{b}\gamma_{\mu}P_{L}b)(\bar{\tau}\gamma^{\mu}P_{L}\tau),$
(16)

respectively, where $M_{V}$ ($V=W^{\prime},Z^{\prime}$) is the gauge boson
mass. We are not assuming the existence of right-handed neutrinos within the
model. Since the $W^{\prime}$ and $Z^{\prime}$ bosons couple primarily to the
fermions from the third-generation, bounds coming from flavor-changing
neutral-currents are avoided. According to electroweak precision data, it is
required that gauge bosons are (almost) degenerate $M_{W^{\prime}}\simeq
M_{Z^{\prime}}$ Faroughy:2016osc . The NP effects are driven by the mass scale
of the heavy mediators and the size of couplings to the third-generation of
fermions $g_{b}$ and $g_{\tau}$. For simplicity, in further numerical analysis
we will take these couplings to be real.

As it was mentioned above, an important caveat of the vector triplet model is
that the parametric space required for the resolution of the $R(D^{(\ast)})$
anomalies and consistency with $\tau^{+}\tau^{-}$ resonance searches at the
LHC (ATLAS and CMS) necessarily implies a very large $Z^{\prime}$ total decay
width, $\Gamma_{Z^{\prime}}/M_{Z^{\prime}}=(g_{\tau}+3g_{b})/(48\pi)\gtrsim
30\%$ Greljo:2015mma ; Faroughy:2016osc . In Sec. III we will show that
current $b\to c\tau\bar{\nu}_{\tau}$ data suggest that tension with
constraints from ATLAS and CMS is now reduced.

###  II.1 Contribution to the charged-current $b\to c\tau\bar{\nu}_{\tau}$ and
neutral-current $b\bar{b}\to\tau\bar{\tau}$ observables

In the SM framework, the $b\to c\tau\bar{\nu}_{\tau}$ quark level processes
are mediated by a virtual $W$ boson exchange. Within the NP scenarios
discussed above, an extra $W^{\prime}$ boson leads to additional tree-level
effective interactions, therefore, modifying the theoretical predictions for
the observables associated with this charged-current transition. The ratios
$R(M)$ ($M=D,D^{\ast},J/\psi$), and the $D^{\ast}$ and $\tau$ longitudinal
polarizations related with the channel $\bar{B}\to
D^{\ast}\tau\bar{\nu}_{\tau}$ can be parametrized as Gomez:2019xfw

$\displaystyle R(M)$ $\displaystyle=$ $\displaystyle R(M)_{\rm
SM}\big{(}\big{|}1+C_{\rm VLL}^{bc\tau\nu_{\tau}}\big{|}^{2}\big{)},$ (17)
$\displaystyle F_{L}(D^{*})$ $\displaystyle=$ $\displaystyle F_{L}(D^{*})_{\rm
SM}\ r_{D^{\ast}}^{-1}\Big{(}\big{|}1+C_{\rm
VLL}^{bc\tau\nu_{\tau}}\big{|}^{2}\big{)},$ (18) $\displaystyle
P_{\tau}(D^{*})$ $\displaystyle=$ $\displaystyle P_{\tau}(D^{*})_{\rm SM}\
r_{D^{\ast}}^{-1}\Big{(}|1+C_{\rm VLL}^{bc\tau\nu_{\tau}}|^{2}\big{)},$ (19)

respectively, where $r_{D^{\ast}}=R(D^{*})/R(D^{*})_{\rm SM}$ and $C_{\rm
VLL}^{bc\tau\nu_{\tau}}$ is the vector left-left (VLL) Wilson coefficient
associated with the NP vector operators given by

$C_{\rm
VLL}^{bc\tau\nu_{\tau}}=\dfrac{\sqrt{2}}{4G_{F}}\dfrac{g_{b}g_{\tau}}{M_{W^{\prime}}^{2}},$
(20)

with $G_{F}$ the Fermi coupling constant. Similarly, the tauonic decay
$B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau}$ and the ratio $R(X_{c})$ of inclusive
semileptonic $B$ decays are also modified as Gomez:2019xfw ; Kamali:2018bdp

$\displaystyle{\rm BR}(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})$ $\displaystyle=$
$\displaystyle{\rm
BR}(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})_{\text{SM}}\big{(}\big{|}1+C_{\rm
VLL}^{bc\tau\nu_{\tau}}\big{|}^{2}\big{)},$ $\displaystyle R(X_{c})$
$\displaystyle=$ $\displaystyle R(X_{c})_{\rm SM}\Big{(}1+1.147\ \big{|}C_{\rm
VLL}^{bc\tau\nu_{\tau}}\big{|}^{2}\Big{)},$ (22)

respectively.

As concerns neutral-current process $b\bar{b}\to\tau\bar{\tau}$, the leptonic
decay ratio $R_{\Upsilon(nS)}$, Eq.(8), is altered by the vector triplet model
Aloni:2017eny . This ratio can be expressed as Aloni:2017eny

$R_{\Upsilon(nS)}=\frac{(1-4x_{\tau}^{2})^{1/2}}{|A_{V}^{\rm
SM}|^{2}}\Big{[}|A_{V}^{b\tau}|^{2}(1+2x_{\tau}^{2})+|B_{V}^{b\tau}|^{2}(1-4x_{\tau}^{2})\Big{]},$
(23)

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

$\displaystyle A_{V}^{b\tau}$ $\displaystyle=$ $\displaystyle-4\pi\alpha
Q_{b}+\frac{m_{\Upsilon(nS)}^{2}}{4}C_{\rm VLL}^{bb\tau\tau},$ (24)
$\displaystyle B_{V}^{b\tau}$ $\displaystyle=$
$\displaystyle-\frac{m_{\Upsilon(nS)}^{2}}{2}C_{\rm VLL}^{bb\tau\tau},$ (25)

where

$C_{\rm VLL}^{bb\tau\tau}=\frac{g_{b}g_{\tau}}{4M_{W^{\prime}}^{2}}.$ (26)

It is straightforward to see the relation between charged and neutral
coefficients, $C_{\rm VLL}^{bc\tau\nu_{\tau}}=(\sqrt{2}/G_{F})C_{\rm
VLL}^{bb\tau\tau}$. In the next section, we will present a phenomenological
analysis of the parametric space of gauge couplings allowed by $b\to
c\tau\bar{\nu}_{\tau}$ and $b\bar{b}\to\tau\bar{\tau}$ data.

##  III Phenomenological study

Dataset | ($g_{b},g_{\tau}$) | $\chi^{2}_{\rm min}/N_{\rm dof}$ |  $p$-value (%) | $\rm{pull_{SM}}$
---|---|---|---|---
$b\to c\tau\bar{\nu}_{\tau}$ | (2.99,1.54) | 1.04 | 39.0 | 3.72
$b\to c\tau\bar{\nu}_{\tau}$ \+ $R_{\Upsilon}$ old | (3.05,1.52) | 0.79 | 61.3 | 3.75
$b\to c\tau\bar{\nu}_{\tau}$ \+ $R_{\Upsilon}$ with BABAR-20 | (3.27,1.39) | 1.20 | 29.3 | 3.68
$b\to c\tau\bar{\nu}_{\tau}$ \+ $R_{\Upsilon}$ combined | (3.05,1.52) | 1.11 | 35.3 | 3.68
Table 1: BFP values of gauge couplings, $\chi^{2}_{\rm min}/N_{\rm dof}$,
$p$-value, and $\rm{pull_{SM}}$ for different datasets of observables.

To provide a robust phenomenological study we consider all of the charged-
current transition $b\to c\tau\bar{\nu}_{\tau}$ observables, namely, the
ratios $R(D^{(\ast)})$ (HFLAV 2019 averages), $R(J/\psi)$, $R(X_{c})$; the
polarizations $P_{\tau}(D^{\ast}),F_{L}(D^{\ast})$; and the upper limit
BR$(B_{c}^{-}\to\tau^{-}\bar{\nu}_{\tau})<10\%$. We will refer to this set as
the $b\to c\tau\bar{\nu}_{\tau}$ data. On the other hand, regarding the
neutral-current observables $R_{\Upsilon(nS)}$, we will take into account
three different datasets:

  1. 1.

$R_{\Upsilon}$ old data: $R_{\Upsilon(1S)}$ BABAR-10 delAmoSanchez:2010bt ,
$R_{\Upsilon(2S)}$ CLEO-07 Besson:2006gj , and $R_{\Upsilon(3S)}$ CLEO-07
Besson:2006gj .

  2. 2.

$R_{\Upsilon}$ with BABAR-20 data: $R_{\Upsilon(1S)}$ BABAR-10
delAmoSanchez:2010bt , $R_{\Upsilon(2S)}$ CLEO-07 Besson:2006gj , and
$R_{\Upsilon(3S)}$ BABAR-20 Lees:2020kom .

  3. 3.

$R_{\Upsilon}$ combined data: $R_{\Upsilon(1S)}$ BABAR-10 delAmoSanchez:2010bt
, $R_{\Upsilon(2S)}$ CLEO-07 Besson:2006gj , and $R_{\Upsilon(3S)}$ average of
CLEO-07 Besson:2006gj  and BABAR-20 Lees:2020kom ,

with $\Upsilon\equiv\Upsilon(nS)$ for simplicity. The purpose of these sets is
to estimate the impact of the very recent BABAR measurement on
$R_{\Upsilon(3S)}$ Lees:2020kom . Furthermore, we complement this analysis by
exploring two plausible scenarios on the $R(D^{(\ast)})$ future measurements
in the ongoing Belle II experiment Kou:2018nap . The two projected scenarios
are as follows Cardozo:2020uol , Belle II-P1: Belle II measurements on
$R(D^{(\ast)})$ keep the central values of Belle combination averages with the
projected Belle II sensitivities for $50\ {\rm ab}^{-1}$ Kou:2018nap ; and
Belle II-P2: Belle II measurements on $R(D^{(\ast)})$ are in agreement with
the current SM predictions at the $0.1\sigma$ level with the projected Belle
II sensitivities for $50\ {\rm ab}^{-1}$ Kou:2018nap . These Belle II future
implications on a $W^{\prime}$ boson scenario have not been explored so far in
previous works.

Bearing in mind the above-mentioned observables, we perform a standard
$\chi^{2}\equiv\chi^{2}(g_{b},g_{\tau})$ function analysis in order to prove
whether it is possible to adjust the deviations of the SM predictions in the
simplified extra gauge bosons model described in Sec. II. We consider the
experimental correlation value $-0.38$ between $R(D)$ and $R(D^{\ast})$ from
HFLAV Amhis:2019ckw ; HFLAVsummer . We determine the regions in the parameter
space favored by the experimental data.

Figure 1: The $1\sigma$ allowed parameter space in the ($g_{b},g_{\tau}$)
plane for the current $b\to c\tau\bar{\nu}_{\tau}$ data [green region] and (a)
$R_{\Upsilon}$ old data [gray region], (b) $R_{\Upsilon}$ with BABAR-20 data
[yellow region], and (c) $R_{\Upsilon}$ combined data [magenta region], for
$M_{W^{\prime}}=1\ {\rm TeV}$. The projection Belle II-P1 (Belle II-P2) for an
integrated luminosity of $50\ {\rm ab}^{-1}$ is represented by the blue (red)
hatched region. The inner black contour lines illustrate the permitted regions
from LHC bounds (solid line) and HL-LHC prospects (dotted line).

###  III.1 Parametric space ($g_{b},g_{\tau}$)

After fitting different sets of observables, we display in Table 1 our results
of the best-fit point (BFP) values on the gauge couplings ($g_{b},g_{\tau}$),
the ratio of the minimum of the $\chi^{2}$ function and number of degrees of
freedom ($\chi^{2}_{\rm min}/N_{\rm dof}$), the $p$-value, and the pull of the
SM $\rm{pull_{SM}}=\sqrt{\chi^{2}_{\rm SM}-\chi^{2}_{\rm min}}$, with
$\chi^{2}_{\rm SM}=\chi^{2}(0)$. In order to keep the couplings in the
perturbative regime ($\sim\sqrt{4\pi}$), we took a benchmark $W^{\prime}$ mass
value of $M_{W^{\prime}}=1$ TeV in our analysis. There is no tension with the
current LHC constraints for the $M_{W^{\prime}}$ (which are above 4 TeV) since
we are assuming zero couplings to the first and second families. For a
$W^{\prime}$ dominantly coupled to a third family, a $W^{\prime}$ mass value
of $\sim 1$ TeV is compatible with LHC bounds (see, for instance, Refs.
Iguro:2020keo ; Marzocca:2020ueu ; Hayreter:2019dzc .) From Table 1, it is
observed that with only $b\to c\tau\bar{\nu}_{\tau}$ data is a good fit
obtained, as expected, with a $p$-value $=39\%$. When $b\to
c\tau\bar{\nu}_{\tau}$ and $R_{\Upsilon}$ old data are joined together, a
better fit is obtained with a larger $p$-value of $61.3\%$. This indicates
that the extra gauge bosons model can simultaneously explain both charged-
current and neutral-current data of $b$-flavored mesons. However, once the
BABAR measurement on $R_{\Upsilon(3S)}$ Lees:2020kom  is incorporated into the
fit, through either $R_{\Upsilon}$ with BABAR-20 or $R_{\Upsilon}$ combined
data, it induces tension in the analysis, causing the quality of the fit to
decrease (smaller $p$-value), but maintaining almost the same value of BFP and
$\rm{pull_{SM}}$. In turn, BABAR’s result Lees:2020kom  seems to challenge
this NP explanation.

In Figs. 1(a), 1(b), and 1(c) we show the $1\sigma$ allowed parameter space in
the ($g_{b},g_{\tau}$) plane, where the gray, yellow, and magenta regions are
obtained by considering $R_{\Upsilon}$ old data, $R_{\Upsilon}$ with BABAR-20
data, and $R_{\Upsilon}$ combined data, respectively. In all of the panels,
the green region represents the allowed region by the charged-current
transition $b\to c\tau\bar{\nu}_{\tau}$ data, and the projection Belle II-P1
(Belle II-P2) for an integrated luminosity of $50\ {\rm ab}^{-1}$ is
represented by the blue (red) hatched region. To further extend our analysis,
the inner black contour lines illustrate the permitted regions from LHC bounds
(solid line) and the prospects at the high-luminosity (HL)-LHC (dotted line)
Iguro:2020keo ; Marzocca:2020ueu . 222These contours have been obtained by
taking into account the LHC bounds on the left-handed vector WC of $|C_{\rm
VLL}^{cb\tau\nu_{\tau}}|\simeq 0.3$ and the future prospects values at HL-LHC
of $|C_{\rm VLL}^{cb\tau\nu_{\tau}}|\simeq 0.1$, evaluated at 1 TeV scale
Iguro:2020keo ; Marzocca:2020ueu . From Fig. 1(a) one can note that it is
possible to get an allowed region on the parameter space to account for a
joint explanation to the $b\to c\tau\bar{\nu}_{\tau}$ and $R_{\Upsilon}$ old
data, in consistency with LHC and HL-LHC bounds. As for Figs. 1(b) and 1(c),
the datasets $R_{\Upsilon}$ with BABAR-20 and $R_{\Upsilon}$ combined prove a
different parametric space not compatible with $b\to c\tau\bar{\nu}_{\tau}$
data. Thus, we confirm that the recent BABAR results on $R_{\Upsilon(3S)}$
generates tension; therefore, charged-current and neutral-current data of
$b$-flavored mesons cannot be addressed simultaneously in this model. Only
relaxing the $R_{\Upsilon(3S)}$ experimental uncertainties to the $2\sigma$
level can a common allowed region be obtained. Regarding the Belle II
experiment, the projection Belle II-P2 indicates that the parametric space
would be severely constrained, but still allow a window for significant NP
contributions. Remarkably, the Belle II-P2 scenario would provide stronger
bounds on the ($g_{b},g_{\tau}$) plane than prospects at the HL-LHC.

On the other hand, as concerns the vector triplet model interpretation to the
most recent $b\to c\tau\bar{\nu}_{\tau}$ data and its consistency with LHC
searches for $Z^{\prime}$ resonances decaying to $\tau^{+}\tau^{-}$
Greljo:2015mma ; Faroughy:2016osc , it is observed that using the BFP gauge
couplings values we get $\Gamma_{Z^{\prime}}/M_{Z^{\prime}}\simeq 20\%$ for
$M_{Z^{\prime}}=1$ TeV, implying a decrease in the tension. In addition,
future Belle II sensitivity on $R(D^{(\ast)})$ would point to smaller widths
$\Gamma_{Z^{\prime}}/M_{Z^{\prime}}\simeq(1-5)\%$ within the allowed regions
by LHC bounds Greljo:2015mma ; Faroughy:2016osc .

In summary, the recent BABAR results on $R_{\Upsilon(3S)}$ hint toward a new
anomalous measurement. Here, we exemplified its implications on the vector
triplet model. Because in a typical electroweak extension of the SM the
charged current parameters and the corresponding ones in the neutral sector
are related, this analysis can be carried out in most of the models involving
the structure
$(\bar{c}\gamma_{\mu}P_{L}b)(\bar{\tau}\gamma^{\mu}P_{L}\nu_{\tau})$. Thus,
our conclusions can be extrapolated to those models. Alternative NP scenarios
can give rise to the same left-left vector operator such as vector leptoquark
models; therefore, it is interesting to explore the possible effects of
$R_{\Upsilon(3S)}$ on these scenarios Garcia:2021 .

##  IV Concluding remarks

New physics scenarios aiming to provide an explanation to the LFU violation
anomalies reported in the charged-current observables of semileptonic $B$
meson decays also induce effects in the neutral-current observables of
bottomonium mesons $R_{\Upsilon(nS)}$, with $n=1,2,3$. Motivated by the very
recent BABAR measurement on $R_{\Upsilon(3S)}$, we revisited the simplified
scenario of extra massive gauge bosons ($W^{\prime}$ and $Z^{\prime}$) that
coupled predominantly to leptons of the third generation (involving LH
neutrinos), proposed as a viable solution to the $b\to c\tau\bar{\nu}_{\tau}$
anomalies. We performed a robust phenomenological analysis of the parametric
space of gauge couplings allowed by the most recent charged-current $b\to
c\tau\bar{\nu}_{\tau}$ and neutral-current $b\bar{b}\to\tau\bar{\tau}$ data.
As the main result of our analysis, it is found that the BABAR measurement of
$R_{\Upsilon(3S)}$ is particularly challenging and the $1\sigma$ range
uncertainties cannot be explained simultaneously with charged-current $b\to
c\tau\bar{\nu}_{\tau}$ data within the LH vector bosons model. Therefore, this
NP scenario seems to be disfavored by BABAR data. In order to clarify this
situation, future $R_{\Upsilon(3S)}$ measurements in the ongoing experiments
Belle II and LHCb will be a matter of importance to confirm or refute the
discrepancy.

###### Acknowledgements.

J. H. Muñoz is grateful to Oficina de Investigaciones of Universidad del
Tolima for financial support of Project No. 290130517. The work of N. Quintero
has been financially supported by MINCIENCIAS and Universidad del Tolima
through Convocatoria Estancias Postdoctorales No. 848-2019 (Contract No.
834-2020). E. Rojas acknowledges financial support from the “Vicerrectoría de
Investigaciones e Interacción Social VIIS de la Universidad de Nariño”,
Projects No. 1928 and No. 2172.

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