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

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 → sµ + µ − and b → cτ ¯ ν τ 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 ( ∗ ) ) = BR( B → D ( ∗ ) τ ¯ ν τ ) / BR( B → D ( ∗ ) (cid:96) (cid:48) ¯ ν (cid:96) (cid:48) ), R (Λ c ) = BR(Λ b → Λ c τ ¯ ν τ ) / BR(Λ b → Λ c µ ¯ ν µ ), and R K ( ∗ ) = BR( B → K ( ∗ ) µ + µ − ) / BR( B → K ( ∗ ) 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 ( ∗ ) ), 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 ( ∗ ) ). 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.

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. [64] 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 modeldependent 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.[55,56,59,60] 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.

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 [55][56][57][58][59][60][61][62][63][64] 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 where, V µ stands for the extra or new vector bosons that transform as (1, 3, 0) under the SU (3) C ⊗ SU (2) L ⊗ U (1) Y gauge symmetry and must be redefined as W ± , Z .On the other side, SM fermions are arranged into the doublets Ψ Q L and Ψ L given by 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 = ΨQ iL γ µ σ I Ψ Q jL and J = Ψ iL γ µ σ I Ψ jL , the effective lagrangian is therefore The middle term of the right-hand side of the above equation corresponds to Substituting equation (12) in the last expression, it leads us to 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 and Z introduced in this model and which masses can naively be considered to be (almost) degenerated which is required by electroweak precision data [58].For simplicity, and without losing generality, we are going to consider that the couplings g q, 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 [55][56][57][58][59][60][61][62].Lastly, the resultant compact form for the couplings of the quark sector to the W that we obtained are where α 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 boson.In this case we find two possibilities: one on flavor conserving interaction given by the other is related to flavor changing Z couplings mediated by where α = β 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 interactions ruled by the lagrangian A. 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 [55,60], 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 These matrices will leave an explicit dependence of these mixing angles (θ 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 [55], and we can establish the full equivalence between the notations of the angles by the relations θ D = α sb and θ L = α µτ .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 thus, we obtain the equivalence and for the leptonic sector The comparison and equivalence among parameterizations of different influential references can be found in Tables II, III, IV and V.
For our last comparison, we considered the parameterization given in Refs.[56,59] 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 λ (q, ) ij .This incidence of the flavor structure into the model can be shown using the charged effective lagrangian as we did before to obtain the desired dominance of couplings to the second and third families using the flavor matrices mentioned before, the λ ij belonging to the first family must be set to zero.Additionally, the values for λ bb = λ τ τ = 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 II, III, IV and V.  [61] Parameterization in [55,60] Parameterization in [56,59] Parameterization in [61] Parameterization in [55,60] Parameterization in [56,59]  [61] Parameterization in [55,60] Parameterization in [56,59] We make emphasis that the results presented in tables II, III, IV, and V 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 Υ(1S) → µ ± τ ∓ , B → K * µ ± τ ∓ , and τ → µφ).
The W boson leads to additional tree-level contribution to b → c − ν transitions involving leptons from secondand third-generation ( = µ, τ ).The total low-energy effective Lagrangian has the following form [66] where G F is the Fermi coupling constant, V cb , is the charm-bottom Cabbibo-Kobayashi-Maskawa (CKM) matrix element, and C bc ν V is the Wilson coefficient (WC) associated with the NP vector (left-left) operator.This WC is defined as with M V the heavy boson mass.The NP effects on the LFU ratios R(X) (X = D, D * , J/ψ), the D * and τ longitudinal polarizations related with the channel B → D * τ ντ , the ratio of inclusive decays R(X c ), and the tauonic decay respectively, where r D * = R(D * )/R(D * ) SM .For BR(B − c → τ − ντ ), we will use the bound < 10% [49].Concerning to the ratio R(Λ c ) very recently measured by LHCb [51], the SM contribution is also rescaled by the overall factor A long term integrated luminosity of 50 ab −1 is expected to be accumulated by the Belle II experiment [68], allowing improvements at the level of ∼ 3% and ∼ 2% for the statistical and systematic uncertainties of R(D) and R(D * ), respectively [68].It is also envisioned accuracy improvements on angular analysis in B → D * τ ντ decay (τ polarization observable P τ (D * )), as well as on q 2 -distribution [68].On the other hand, the LHCb will be able to improve measurements of R(D * ) and R(J/ψ) in the future runs of data taking [2,3].
In regard to the transition b → cµν µ , the µ/e LFU ratios R  I).The W boson coupling to lepton pair µν µ modifies this ratio as where is given by Eq. (27).From this LFU ratio we get the bound which is relevant for the couplings aiming to explain the b → sµ + µ − anomaly (see Sec. III C).
The TVB model can also induce NP contributions in the leptonic decay B → ν induced via the charged-current transition b → u − ν ( = µ, τ ).The ratio provides a clean LFU test [71].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 [73]).The NP effects on this ratio can be expressed as where and The experimental value is [R τ /µ B ] Exp = 205.7 ± 96.6, which was obtained from the values reported by the Particle Data Group (PDG) on BR(B − → τ − ντ ) [74] and the Belle experiment on BR(B − → µ − νµ ) [75].
The NP effective Lagrangian responsible for the semileptonic transition b → sµ + µ − can be expressed as where the NP is encoded in the WCs C bsµµ respectively, with α em being the fine-constant structure.A global fit analysis including most current b → sµ + µ − data, such as R K ( * ) by LHCb [9,10] and BR(B s → µ + µ − ) by CMS [11], has been recently performed in Ref. [26,27].Among the different NP scenarios, the solution is preferred by the data [26,27]. 2The best fit 1σ solution is [26] In the context of the TVB model, the Z boson induces a tree-level contribution to b → sµ + µ − transition via the WCs Using the result of the global fit, Eq. ( 43), this corresponds to 2 Let us notice that the single WC C bsµµ 9 also provides a good fit of the b → sµ + µ − data [26,27].Some explicit model examples are shown in [26].
The NP effects of the TVB model on the leptonic ratio can be expressed as [76,77] with The neutral gauge boson also generates the LFV processes Υ → µ ± τ ∓ (Υ ≡ Υ(nS)).The branching fraction is given by [60,61] where f Υ and m Υ are the Upsilon decay constant and mass, respectively.The decay constant values can be extracted from the experimental branching ratio measurements of the processes Υ → e − e + .Using current data from PDG [74], one obtains f Υ(1S) = (659 ± 17) MeV, f Υ(2S) = (468 ± 27) MeV, and f Υ(3S) = (405 ± 26) MeV.Experimentally, the reported ULs are BR(Υ(1S) → µ ± τ ∓ ) < 2.7 × 10 −6 from Belle [82], and BR(Υ(2S) [74].From these ULs we get E. ∆F = 2 processes: Bs − Bs and D 0 − D0 mixing The interactions of a Z boson to quarks s b relevant for b → sµ + µ − processes also generate a contribution to B s − Bs mixing [83,84].The NP effects to the B s − Bs mixing can be described by the effective Lagrangian where Thus, the NP contributions to the mass difference ∆M s of the neutral B s meson can be expressed as [83] ∆M SM+NP s where η = α s (M Z )/α s (m b ) accounts for running from the M Z scale down to the b-quark mass scale and the SM loop function is R loop SM = (1.310± 0.010) × 10 −3 [83].At present, ∆M s has been experimentally measured with great precision ∆M Exp s = (17.757± 0.021) ps −1 [41,83].On the theoretical side, the average is ∆M SM s = (18.4+0.7 −1.2 ) ps −1 implying that ∆M SM s /∆M Exp s = 1.04 +0.04 −0.07 [83].This value yields to where in the TVB model translates into the important 2σ bound In addition, the Z boson can also admit c → u transitions, consequently generating tree-level effects on D 0 − D0 mixing [61,85].The effective Lagrangian describing the Z contribution to D 0 − D0 mixing can be expressed as [61,85] where g uc = g q bb V cb V * ub + g q sb (V cs V * ub + V cb V * us ) + g q ss V cs V * us [61] (see also Table IV).Such a NP contributions are constrained by the results of the mass difference ∆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 [61,85].Here we follow the recent analysis of Ref. [61] focused on short-distance SM contribution that sets the conservative (strong) bound The couplings g q bb and g q sb are less constrained by ∆M D [61], therefore, we will skip them in our study.

F. Neutrino Trident Production
The Z couplings to leptons from second-generation (g µµ = g νµνµ ) also generate a contribution to the cross-section of neutrino trident production (NTP), ν µ N → ν µ N µ + µ − [86].The cross-section is given by [86] where v = ( √ 2G F ) −1/2 and s W ≡ sin θ W (with θ W the Weinberg angle).The existing CCFR trident measurement σ CCFR /σ SM = 0.82 ± 0.28 provides the upper bound G. LFV B decays: The current experimental limits (90% C.L.) on the branching ratios of Let us notice that LHCb Collaboration obtained a limit of BR(B + → K + µ − τ + ) LHCb < 3.9 × 10 −5 [87] 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 → K * 0 µ ± τ ∓ [88].The obtained UL on this LFV decay is [88] BR From the theoretical side, the branching ratio of 55] can be written as respectively, where (a K , b K ) = (12.72 ± 0.81, 13.21 ± 0.81) [89], and (a 1, 15.4±1.9)[55] are the numerical coefficients that have been calculated using the B → K ( * ) transitions form factors obtained from lattice QCD [55,89].The decay channel with final state µ − τ + can be easily obtained by replacing µ τ .The current ULs can be translated into the bounds As for the LFV leptonic decay B s → µ ± τ ∓ , the branching ratio is [55] BR where f Bs = (230.3± 1.3) MeV is the B s decay constant [41] and we have used the limit m τ m µ .Recently, the LHCb experiment has reported the first upper limit of BR(B s → µ ± τ ∓ ) < 4.2 × 10 −5 at 95% CL [90].Thus, one gets the following limit H. Rare B decays: Recently, the interplay between the di-neutrino channel B → K ( * ) ν ν and the B meson anomalies has been studied by several works [85,[91][92][93][94].In the NP scenario under study, the Z boson can give rise to B → K ( * ) ν ν at tree level.The effective Hamiltonian for the b → sν ν transition is given by [95] where L is the aggregate of the SM contribution C SM L ≈ −6.4 and the NP effects ∆C ij L , that in the TVB framework read as with i, j = µ, τ .By defining the ratio [95] the NP contributions can be constrained.In the TVB model this ratio is modified as From this expression, we can observe that diagonal leptonic couplings g µµ and g τ τ contribute to b → sν µ νµ (relevant for b → sµ + µ − data) and b → sν τ ντ (relevant for b → cτ ντ 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 → sν µ ντ (and ν τ νµ ) through the off-diagonal coupling g µτ .

J. LHC bounds
LHC constraints are always important for models with non-zero Z couplings to the SM particles [105].In particular, in our study it will set important constraints on the parametric space conformed by the TVB couplings (g q bb , g µµ ) and (g q bb , g τ τ ).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 √ s = 13 TeV during Run 2 of the LHC with an integrated luminosity of 139 fb −1 [106] (recently, the CMS collaboration has also reported constraints for similar luminosities [107], basically identical to ATLAS [106]), and the data from searches of Z bosons decaying to tau pairs with an integrated luminosity of 36.1 fb −1 from proton-proton collisions at √ s = 13 TeV [108].There are also searches for high-mass resonances in the monolepton channels (pp → ν) carried out by the ATLAS and CMS [109][110][111].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 [106,108] with the theoretical cross-section given in Ref. [112].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 production processes in the b b annihilation and the subsequent Z decay into muons (µ + µ − ) and taus (τ + τ − ).Further details are shown in Refs.[112][113][114].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 → sµ + µ − and b → cτ ντ data.We define the pull for the i-th observable as where O exp i is the experimental measurement, O th i ≡ O th i (g q bs , g q bb , g µµ , g τ τ , g µτ ) is the theoretical prediction that include the NP contributions, and ∆O i = ((σ exp i ) 2 + (σ 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 χ 2 function is written as the sum of squared pulls, i.e., where the sum extends over the number of observables (N 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 → cτ ντ and b → sµ + µ − data, bottomonium ratios R Υ(nS) , LFV decays (B , ∆F = 2 processes, and neutrino trident production.We will study the impact of the most recent LHCb measurements on the ratios R(D ( * ) ) [42][43][44], 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) LHCb22 + R(D * ) LHCb23 , • All data with R(D ( * ) ) LHCb22 , • All data with R(D ( * ) ) HFLAV23 .
All these three sets contain a total number of observables N obs = 31 and five free TVB parameters (g q bs , g q bb , g µµ , g τ τ , g µτ ) to be fitted.The heavy TVB mass will be fixed to the benchmark value M V = 1 TeV.Therefore, the number of degrees of freedom is N dof = 26.
For the three sets of observables we find the best-fit point values by minimizing the χ 2 function (χ 2 min ).In Table VI we report our results of the best-fit point values and 1σ intervals of TVB couplings.For each fit we also present in Table VI the values of χ 2 min /N 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 | ∼ O(10 −3 ), and opposite sign to g µµ to be consistent with b → sµ + µ − data (C µµ 9 = −C µµ 10 solution) and B s − Bs mixing.On the other hand, large values for the bottom-bottom coupling g q bb ∼ O(1) are preferred.As for the leptonic couplings, it is found that the lepton flavor conserving ones have a similar size g µµ ≈ g τ τ ∼ O(10 −1 ) for All data with R(D) LHCb22 + R(D * ) LHCb23 , suggesting non-hierarchy pattern.While for All data with R(D ( * ) ) LHCb23 (with R(D ( * ) ) HFLAV23 ), these couplings exhibit a hierarchy g τ τ > g µµ .As LFV coupling concerns, the obtained best-fit point values on g µτ 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 TeV).We obtain that in order to avoid large values on g q bb coupling (∼ √ 4π), that would put the perturbativity of the model into question, the TVB mass can be as large as M V ∼ 2 TeV.
In Fig. 1, we show the allowed regions of the most relevant two-dimension (2D) parametric space of (a) All data with R(D) LHCb22 + R(D * ) LHCb23 , (b) All data with R(D ( * ) ) LHCb22 , and (c) All data with R(D ( * ) ) HFLAV23 , respectively, for a benchmark TVB mass M V = 1 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 [106,108], as discussed in Sec.III J.For completeness, the perturbative region (g q bb ≥ √ 4π)) is represented by yellow color.It is observed in the planes (g q bb , g τ τ ) and (g q bb , g µµ ) for All data with R(D ( * ) ) HFLAV23 that the TVB model is seems to be strongly ruled out by the LHC bounds.However, for All data with R(D) LHCb22 + R(D * ) LHCb23 (and with R(D ( * ) ) LHCb22 ) that include the very recent LHCb measurements [42][43][44], the TVB model can provide a combined explanation of the b → cτ ντ and b → sµ + µ − 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 → cτ ντ 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 → cτ ντ and b → sµ + µ − data [61].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 ) [61].Such a result is in agreement with ours for All data with R(D ( * ) ) 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 [76] where only the charged-current b → cτ ντ anomaly was addressed within this framework.

A. 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 [55,60], as it was previously discussed in Sec.II, we found that the equivalence in the quark sector is Eq. ( 23), while for the leptonic sector we have Eq.(24).Taking into account the 1σ range solutions of TVB couplings obtained in Table VI (for the three sets of data), we V cb ) result is still in agreement with previous analysis [55,60].On the contrary, in the leptonic sector, we obtained that because of 1σ range of the LFV coupling g µτ it is not possible to find a physical solution to the mixing angle θ L .As additional probe, we have performed a global fit to the current b → sµ + µ − and b → cτ ντ data, and the most relevant flavor observables, with (g q 2 , g 2 , θ D , θ L ) as free parameters.For a fixed mass value M V = 1 TeV, we obtained a very poor fit (χ 2 min /N dof 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 → cτ ντ and b → sµ + µ − 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 ( * ) ) and R(Λ c ).As concerns b → sµ + µ − data, we taken into account the C bsµµ 9 = −C bsµµ 10 solution from global fit analysis including the recent results on R K ( * ) by LHCb and BR(B s → µ + µ − ) by CMS.We have also included all relevant flavor observables such as B s − Bs mixing, neutrino trident production, LFV decays (B → K ( * ) µ ± τ ∓ , B s → µ ± τ ∓ , τ → µφ, Υ(nS) → µ ± τ ∓ ), rare B decays (B → K ( * ) ν ν, B → Kτ + τ − , B s → τ + τ − ), 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 ( * ) ) reported by HFLAV, the TVB model can accommodate the b → cτ ντ and b → sµ + µ − 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 ( * ) ).The the B meson anomalies can be addressed within the TVB model in consistency with LHC constraints.We concluded that new and improved b → cτ ντ 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.

9 and C bsµµ 10 of
the four-fermion operators FIG. 1. 68% (green) and 95% (light-green) CL allowed regions for the most relevant 2D parametric space of (a) All data with R(D) LHCb22 + R(D * ) LHCb23 , (b) All data with R(D ( * ) ) LHCb22 , and (c) All data with R(D ( * ) )HFLAV23, respectively, for MV = 1 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 ≥ √ 4π)) is represented by yellow color.

TABLE I .
Experimental status and SM predictions on observables related to the charged-current transitions b → c ν ( = µ, τ ).

TABLE II .
Couplings to W boson in different parameterizations of the TVB model Coupling Parameterization in

TABLE III
. Flavor conserving couplings to Z boson in different parameterizations of the TVB model.Coupling

TABLE IV
. Flavor changing couplings to Z boson in different parameterizations of the TVB model.Coupling Parameterization in

TABLE V .
Couplings of leptons to Z boson in different parameterizations of the TVB model.

TABLE VI .
Best-fit point values and 1σ intervals of the five TVB couplings (g q bs , g q bb , g µµ , g τ τ , g µτ ) for the three different sets of observables and a benchmark mass value of MV = 1 TeV.