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

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 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 be easily accommodated (within its experimental $1\sigma$ range) together with the other $b \to c\tau\bar{\nu}_\tau$ data, hinting towards a new anomalous observable.

Lepton flavor universality can be tested through the ratio of semileptonic B meson decays and leptonic Υ meson decays, with Υ ≡ Υ(nS) (n = 1, 2, 3). For the charged-current transitions b → cτντ , discrepancies between the experiment and the Standard Model (SM) have been observed in recent years by different flavor facilities as BABAR, Belle, and LHCb. While for the neutralcurrent transitions bb → ττ , the BABAR experiment reported recently a new measurement of leptonic decay ratio R Υ(3S) = BR(Υ(3S) → τ + τ − )/BR(Υ(3S) → µ + µ − ), showing an agreement with the SM at the 1.8σ level. In light of this new BABAR result and regarding the connection between new physics (NP) interpretations to the charged-current b → cτντ anomalies and neutralcurrent bb → ττ 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 Υ(3S) cannot be easily accommodated (within its experimental 1σ range) together with the other b → cτντ data, hinting towards a new anomalous observable.
On the other hand, LFU can be also tested through the ratio of leptonic decays of bottomonium meson Υ(nS) [61] R Υ(3S) = CLEO-07: 1.05 ± 0.08 ± 0.05 [63], SM: 0.9948 [61], (11) where the theoretical uncertainty is tipically of the order ±O(10 −5 ) [61]. These measurements are in good accordance with the SM by 0.5σ, 0.8σ, and 0.6σ, respectively. Recently, in 2020 the BABAR experiment has released a new measurement on the ratio R Υ(3S) [64], whose value is which improves the precision of the experimental value previously obtained by CLEO [63]. Despite this improvement, the new value is below the SM expectation and shows an agreement at the 1.8σ level [64], in higher tension than CLEO. Moreover, averaging the CLEO-07 [63] and BABAR-20 [64] measurements we obtain which deviates at the 1.7σ level with respect to the SM prediction (uncertainties were taken in quadrature). Motivated by the tension generated by the new BABAR measurement on R Υ(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 ( * ) ) anomalies also induce inevitable effects in the leptonic decay ratio R Υ(nS) [61]. The connection between chargedcurrent b → cτν τ and neutral-current bb → ττ processes was first pointed out by the authors of Ref. [38], in which they performed a recast of existing τ + τ − resonance searches at the CMS and ATLAS experiments, allowing to set constraints on different simplified models addressing the R(D ( * ) ) anomalies. Keeping in mind the correlation between NP solutions to the charged-current b → cτν τ anomalies and neutral-current bb → ττ processes [38,61], and to the light of the very recent BABAR result on R Υ(3S) [64], in this work we present a reanalysis of the left-handed vector bosons model that preferentially couples to thirdgeneration fermions (also referred to as vector triplet model) [37,38]. As it was above mentioned, this model remains as a viable explanation to the b → cτν τ anomalies. A previous analysis addressing the R(D ( * ) ) anomalies and the complementary of R Υ(nS) in this model was presented in Ref. [61], in which the authors found within 95% confidence level the numerical values for the Wilson coefficients which minimize the observed anomaly in R(D ( * ) ), and the corresponding predictions for R Υ(nS) . This study was implemented by considering the 2016 HFLAV averages [65], which differ from the most recent 2019 HFLAV ones [12,13]. 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 → cτν τ and R Υ(nS) data. Particularly, for the b → cτν τ data, we include the polarizations of D * and the tau lepton, R(J/ψ), the upper limit on BR(B − c → τ −ν τ ), and incorporate the forthcoming sensitivity of Belle II on R(D ( * ) ) measurements. In that sense, our work complements and extends the previous analysis performed in [61]. We will show that the vector triplet model is in conflict with the BABAR measurement of R Υ(3S) and the 1σ range uncertainties cannot be explained in simultaneity with b → cτν τ data.
The outline of this paper is organized as follows. In Sec. II, we briefly present the main features of the lefthanded 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 and Z that coupled predominantly to left-handed (LH) fermions from the third-generation [37,38]. The Lagrangian describing the interactions between fermions and vector boson is [37,38] where Q 3 = (V cb c L , b L ) T and L 3 = (ν τ L , τ L ) T are the LH quark and lepton doublets, σ 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 τ the cor-responding gauge couplings to quarks and leptons, 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 → cτν τ and neutral-current bb → ττ operators are given by [37] respectively, where M V (V = W , Z ) is the gauge boson mass. We are not assuming the existence of right-handed neutrinos within the model. According to electroweak precision data, it is required that gauge bosons are (almost) degenerate M W M Z [38]. The NP effects are driven by the mass scale of the extra gauge bosons and the size of couplings to the third-generation of fermions g b and g τ . For simplicity, in further numerical analysis we will take these couplings to be real.
A. Contribution to the charged-current b → cτντ and neutral-current bb → ττ observables In the SM framework, the b → cτν τ quark level processes are mediated by a virtual W boson exchange. Within the NP scenarios discussed above, an extra W 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 * , J/ψ), and the D * and τ longitudinal polarizations related with the channel B → D * τν τ can be parametrized as [47] respectively, where r D * = R(D * )/R(D * ) SM and C bcτ ντ VLL is the vector left-left (VLL) Wilson coefficient associated with the NP vector operators given by with G F the Fermi coupling constant. Similarly, the tauonic decay B − c → τ −ν τ and the ratio R(X c ) of inclusive semileptonic B decays are also modified as [47,66] R(X c ) = R(X c ) SM 1 + 1.147 C bcτ ντ respectively.
As concerns neutral-current process bb → ττ , the leptonic decay ratio R Υ(nS) , Eq.(8), is altered by the vector triplet model [61]. This ratio can be expressed as [61] where It is straightforward to see the relation between charged and neutral coefficients, C bcτ ντ VLL = ( In the next section, we will present a phenomenological analysis of the parametric space of gauge couplings allowed by b → cτν τ and bb → ττ data.
3. R Υ combined data: R Υ(1S) BABAR-10 [62], R Υ(2S) CLEO-07 [63], and R Υ(3S) average of CLEO-07 [63] and BABAR-20 [64], with Υ ≡ Υ(nS) for simplicity. The purpose of these sets is to estimate the impact of the very recent BABAR measurement on R Υ(3S) [64]. Furthermore, we complement this analysis by exploring two plausible scenarios on the R(D ( * ) ) future measurements in the ongoing Belle II experiment [67]. The two projected scenarios are as follows [68], Belle II-P1: Belle II measurements on R(D ( * ) ) keep the central values of Belle combination averages with the projected Belle II sensitivities for 50 ab −1 [67]; and Belle II-P2: Belle II measurements on R(D ( * ) ) are in agreement with the current SM predictions at the 0.1σ level with the projected Belle II sensitivities for 50 ab −1 [67]. These Belle II future implications on a W boson scenario have not been explored so far in previous works.
Bearing in mind the above-mentioned observables, we perform a standard χ 2 ≡ χ 2 (g b , g τ ) 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 * ) from HFLAV [12,13]. We determine the regions in the parameter space favored by the experimental data. After fitting different sets of observables, we display in Table I our results of the best-fit point (BFP) values on the gauge couplings (g b , g τ ), the ratio of the minimum of the χ 2 function and number of degrees of freedom (χ 2 min /N dof ), the p-value, and the pull of the SM pull SM = χ 2 SM − χ 2 min , with χ 2 SM = χ 2 (0). In order to keep the couplings in the perturbative regime (∼ √ 4π), we took a benchmark W mass value of M W = 1 TeV in our analysis. There is no tension with the current LHC constraints for the M W (which are above 4 TeV) since we are assuming zero couplings to the first and second families. For a W dominantly coupled to third family, a W mass value of ∼ 1 TeV is compatible with LHC bounds (see, for instance, Refs. [22,49,50].) From Table I, it is observed that with only b → cτν τ data a good fit is obtained, as expected, with a p-value = 39%. When b → cτν τ and R Υ 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 Υ(3S) [64] is incorporated into the fit, either through R Υ with BABAR-20 or R Υ combined data, 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 pull SM . In turn, BABAR's result [64] seems to challenge this NP explanation.
In Figs. 1(a), 1(b), and 1(c) we show the 1σ allowed parameter space in the (g b , g τ ) plane, where the gray, yellow, and magenta regions are obtained by considering R Υ old data, R Υ with BABAR-20 data, and R Υ combined data, respectively. In all of the panels, the green region represents the allowed region by the charged-current transition b → cτν τ data, and the projection Belle II-P1 (Belle II-P2) for an integrated luminosity of 50 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) [22,49] 1 . From Fig. 1(a) one can note that is possible to get an allowed region on the parameter space to account for a joint explanation to the b → cτν τ and R Υ old data. As for the Figs. 1(b) and 1(c), the data sets R Υ with BABAR-20 and R Υ combined prove a different parametric space not compatible with b → cτν τ data. Thus, we confirm that the recent BABAR results on R Υ(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 Υ(3S) experimental uncertainties to the 2σ level a common allowed region can be obtained. Regarding the Belle II experiment, the projection Belle II-P2 indicates that the parametric space would be severely constrained, but still allowing a window for significant NP contributions. Remarkably, Belle II-P2 scenario would provide stronger bounds on the (g b , g τ ) plane than prospects at the HL-LHC.
The recent BABAR results on R Υ(3S) hinting towards a new anomalous measurement. Here, we exemplified its implications on the LH vector bosons model. Alternative NP scenarios can give rise to the vector operator (cγ µ P L b)(τ γ µ P L ν τ ), such as vector leptoquark models, thus, it is interesting to explore the possible effects of R Υ(3S) on these scenarios [69].

IV. CONCLUDING REMARKS
New physics scenarios aiming to provide an explanation to the anomalies reported in the charged-current observables of semileptonic B meson decays, also induce effects in the neutral-current observables of bottomonium mesons R Υ(nS) , with n = 1, 2, 3. Motivated by the very recent BABAR measurement on R Υ(3S) , we revisited the simplified scenario of extra massive gauge 1 These contours have been obtained by taking into account the LHC bounds on the left-handed vector WC of |C cbτ ντ VLL | 0.3 and the future prospects values at HL-LHC of |C cbτ ντ VLL | 0.1, evaluated at 1 TeV scale [22,49]. bosons (W and Z ) that coupled predominantly to leptons of the third generation (involving LH neutrinos), proposed as a viable solution to the b → cτν τ anomalies. We performed a robust phenomenological analysis of the parametric space of gauge couplings allowed by charged-current and neutral-current data. As the main result of our analysis, it is found that the BABAR measurement of R Υ(3S) is particularly challenging and the 1σ range uncertainties cannot be explained simultaneously with charged-current b → cτν τ 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 Υ(3S) measurements in the ongoing experiments Belle II and LHCb will be a matter of importance to confirm or refute the discrepancy.