On the breaking of Lepton Flavor Universality in B decays

In view of recent experimental indications of violations of Lepton Flavor Universality (LFU) in $B$ decays, we analyze constraints and implications of LFU interactions, both using an effective theory approach, and an explicit dynamical model. We show that a simple dynamical model based on a $SU(2)_L$ triplet of massive vector bosons, coupled predominantly to third generation fermions (both quarks and leptons), can significantly improve the description of present data. In particular, the model decreases the tension between data and SM predictions concerning: i) the breaking of $\tau$-$\mu$ universality in $B\to D^{(*)} \ell \nu$ decays; ii) the breaking of $\mu$-$e$ universality in $B \to K \ell^+\ell^-$ decays; iii) the difference between exclusive and inclusive determinations of $|V_{cb}|$ and $|V_{ub}|$. The minimal version of the model is in tension with ATLAS and CMS direct searches for the new massive vectors (decaying into $\tau^+\tau^-$ pairs), but this tension can be decreased with additional non-standard degrees of freedom. Further predictions of the model both at low- and high-energies, in view of future high-statistics data, are discussed.


Introduction
Recent experimental data in B physics hint toward deviations of Lepton Flavor Universality (LFU) in semi-leptonic decays, both in the case of b → c charged-current transitions, as well as in the case of b → s neutral currents. The statistically most significant results can be summarized as follows: • 3.8σ + 2.0σ deviation of τ / universality ( = µ, e) in b → c transitions, encoded by [1][2][3]:

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In addition to these LFU ratios, whose deviation from unity would clearly signal physics beyond the Standard Model (SM), B-physics data exhibit other tensions with SM expectations in semi-leptonic observables. Most notably, a ∼ 3σ deviation from the SM expectation has been reported by LHCb [9,10] in the so-called P 5 differential observable of B → K * µ + µ − decays [11]. Moreover, in charged current transitions there is a longstanding ∼ 2.5σ discrepancy in the determination of both |V cb | and |V ub | from exclusive vs. inclusive semi-leptonic decays [12].
These deviations from the SM have triggered a series of theoretical speculations about possible New Physics (NP) interpretations, see in particular ref. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Among these recent papers, two particularly interesting observations are: i) the proposal of ref. [20] to explain both R µ/e K and the P 5 anomaly by means of NP coupled dominantly to the third generation of quarks and leptons, with a small non-negligible mixing between third and second generations; ii) the observation of ref. [21] that is natural to establish a connection between R µ/e K and R τ / D * if the effective four-fermion semi-leptonic operators are build in terms of left-handed doublets.
Despite this recent progress, a coherent dynamical picture explaining all the anomalies has not emerged yet. On the one hand, a significantly improved fit of experimental data can be obtained with a specific set of four-fermion operators of the type J q ×J , where J q and J are flavor-non-universal left-handed quark and lepton currents [21,31]. On the other hand, even within an Effective Field Theory (EFT) approach, it is hard to believe that this set of effective operators is the only relevant one in explicit NP models. In particular, explicit NP models should face the tight constraints on four-quark and four-lepton operators dictated by meson-antimeson mixing, and by the bounds on Lepton Flavor Violation (LFV) and LF non-universality in pure leptonic processes. Moreover, the size of the SM modifications in eqs. (1.1)-(1.3) points toward relatively light new degrees of freedom, that could well be within the reach (or already excluded) by direct searches at the LHC.
In this paper we present an attempt to build a simplified coherent dynamical model able to explain, at least in part, these violations of LFU. The guiding principle of our construction is the idea that the J q × J effective operators are generated by the exchange of one set (or more sets) of massive vector bosons that transform as a SU(2) L triplet, and that are coupled to both quark and lepton currents. This hypothesis allows us to establish a connection between quark-lepton, quark-quark, and lepton-lepton effective operators. We further assume that the flavor structure of the new currents is consistent with an approximate U(2) q × U(2) flavor symmetry acting on the first two generations of quarks and leptons, along the lines of ref. [32].
Under these assumptions we proceed with two main steps: i) we analyze the low-energy constraints (and the corresponding phenomenological implications) on the complete set of four-fermion operators generated within the model; ii) we discuss the additional constraints due to electroweak precisions test and collider searches, following from the specific choice of the mediators.
We find that, after taking into account all the existing constraints, the proposed model can still provide a significantly improved fit as far as low-energy observables are concerned. The most serious constraint on the model follows from the searches performed by ATLAS -2 -

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and CMS on new heavy neutral states (Z ) decaying into τ + τ − pairs. However, as we will discuss, the tension with direct searches can be decreased with additional non-standard degrees of freedom, whose net effect is the enhancement of the Z decay width and the corresponding suppression of the Z → τ + τ − branching ratio. The tension can be further reduced in the limit where the assumption of narrow resonances (Γ M ), that is implicit in all present direct searches, no longer holds.

Step I: four-fermion operators
Our main assumption is that all the non-standard four-fermion interactions can be described by the following effective Lagrangian where J a µ is a fermion current transforming as a SU(2) L triplet, built in terms of SM quarks and lepton fields: Here λ q, are Hermitian flavor matrices and, by convention, λ q 33 = λ 33 = 1. We define Q i L and L i L to be the quark and lepton electroweak doublets in the flavor basis where down-type quarks and charged-leptons are diagonal. We assume an approximate U(2) q × U(2) flavor symmetry, under which the light generations of Q i L and L i L transform as 2 q × 1 and 1 q × 2 , respectively, and all other fermions are singlets. We further assume that the underlying dynamics responsible for the effective interaction in eq. (2.1) involves, in first approximation, only third generation SM fermions (the left-handed 1 q ×1 fermions). In this limit, the flavor couplings in eq. (2.2) are λ q, ij = δ i3 δ 3j . The corrections to this limit are expected to be generated by appropriate U(2) q × U(2) breaking spurions, connected to the generation of subleading terms in the Yukawa couplings for the SM light fermions.
In the quark case, the leading U(2) q breaking spurion is a doublet, whose flavor structure is unambiguously connected to the CKM matrix (V ) [32]. We can thus expand λ q ij as follows: . As we will discuss below, low-energy flavor-physics data imply i 1. The breaking structure in the lepton sector is less clear, given the intrinsic ambiguity in reconstructing the lepton Yukawa couplings under the (natural) assumption that neutrino masses are generated by a see-saw mechanism. 2 As we will discuss below, low-energy data are compatible with the hypothesis that the leading breaking terms in the lepton sector transform as doublets of U(2) .

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Among the four-fermion operators generated by the model, the ones most relevant to flavor phenomenology are: Step II: simplified dynamical model In order to generate ∆L 4f in a dynamical way, we introduce the heavy spin-1 triplet, V a µ (a = 1, 2, 3), following the general simplified Lagrangian proposed in ref. [45]. By means of this approach we can describe both models in which the new vector is weakly coupled, such as gauge extension of the SM, and strongly coupled models, such as Composite Higgs models. The simplified Lagrangian reads By integrating out at the tree-level the heavy spin-1 triplet and keeping only effective operators of dimension ≤ 6, we obtain the effective Lagrangian By construction, the first term is ∆L (T ) 4f in eq. (2.1). The second term, in the unitary gauge, is simply This term induces an unphysical (custodially-invariant) shift in the W -and Z-boson masses,. 4 that can be reabsorbed by a redefinition of v, and deviations in the Higgs interactions to W and Z bosons. The latter are well within the existing bounds for the relevant set of parameters. The last term, instead, describes non-universal deviations in the Z and W couplings to SM quarks and leptons that lead to non-trivial constraints on the parameter space of the model (see section 4.1). 3 With respect to ref. [45] we dropped interaction terms with two or more insertions of V a µ . While such terms can be relevant for double production, they do not contribute to the low-energy effective Lagrangian at the dimension-6 level and are thus largely unconstrained by low-energy data. 4 Within the full model of eq. (2.9) this corresponds to a mass mixing between the SM EW gauge bosons and the heavy vector triplet. The relative shift in the heavy vector masses mV is only of

New physics effects in charged currents
Since the new interactions are purely left-handed, in the case of charged currents their effect is simply an overall (flavor non-universal) rescaling of the SM amplitudes: where Using this expression, the LFU breaking ratio in eq. (1.1) assumes the form where we have assumed |λ µµ,ee | 1 and |λ q ij | |V * 3i V 3j |. The first condition is required by the smallness of deviations from the SM in b → c ν decays (see below), the second condition follows by the consistency of the bounds from ∆F = 2 amplitudes (see section 3.2). We are thus able to fix the overall strength of the new effective charged-current interaction (compared to the Fermi coupling): In principle, violations of LFU universality are expected also between b → c(u)µν and b → c(u)eν modes. Unfortunately, it is very difficult to get experimental bounds on the latter using published data, since most of the high-statistics semi-leptonic analyses are performed combining µ and e modes [5]. Using the PDG fit for the combined B ± /B 0 sample [5], that is separated for µ and e modes, we deduce that deviations between Γ(b → c(u)µν) and Γ(b → c(u)eν) as large as ∼ 2% are allowed by present data. Within our model, we expect The strong constraints on LFU involving only quarks and leptons of the first two generations (π and K decays, CKM unitarity, and µ decay [34]) implies |λ ee | |λ µµ |. As a result, the constraints on µ-e charged-current LFU can be used to set the approximate bound that justifies having neglected λ µµ in eq. (3.3). After this bound is imposed, violations of LFU universality in K and π semileptonic decays turn out to be unobservables, given the additional suppression factor |V ub (λ q bq ) * /V uq |, for q = s, d, compared to eq. (3.5). The universal 30% excess to τ semi-leptonic charged-current decays is likely to explain, at least in part, the tension between exclusive and inclusive determinations of |V cb | and |V ub |. The argument goes as follows: the B → X c,u τ ν decays followed by τ → X νν represent a background for the inclusive B → X c,u ν analyses. At present, this background is subtracted via montecarlo simulations that assume a SM-like B(B → X c,u τ ν). 5 This procedure therefore underestimates the background events and leads to an enhanced B → X c,u ν signal. On the other hand, the problem is not present in the exclusive decays of the type B → D ( * ) ν, where the kinematical closure of the events prevents the contamination from τ decays. A precise estimate of the effect would require a re-analysis of B → X c,u ν data and is beyond the scope of the present paper. However, we note that this effect necessarily goes in the right direction (enhanced signal in inclusive modes), and that is likely to be larger in b → u compared to b → c, given the different kinematical cuts. We are then led to the conclusion that the most reliable estimates of |V cb | and |V ub | are those obtained by means of exclusive decays and, more specifically, exclusive decays into electron final states.
As a result of this discussion, we urge the experimental collaborations to reanalyze all semi-leptonic charged-current B decays without imposing LFU, both as far as signal and as far as background are concerned.

Bounds from ∆F = 2
Also in the case of ∆F = 2 transitions the new interaction amounts to an overall flavor non-universal rescaling of the SM amplitudes. It is therefore convenient to define the ratios The consistency with experimental results on down-type ∆F = 2 amplitudes, where no significant deviations from the SM are observed (up to the 10%-30% level depending on the specific amplitude) implies |λ q ij | < ∼ 10 −1 |V * 3i V 3j |, for R 0 = 0.15 and g /g q = O(1). As anticipated, this justifies the expansion on the r.h.s. of eq. (3.3).
If the corrections of λ q ij from the leading term are generated by U(2) q breaking spurions, as proposed in eq. (2.3), the R ∆F =2 Bq terms should respect the U(2) 3 prediction [32] To discuss the NP impact in b → s + − decays we need to establish a precise upper bound on |λ q bs |. According to the U(2) 3 fit of meson-antimeson mixing of ref. [36], we set Even in the limit of negligible λ q ij for i = 3 or j = 3, a potentially sizable contribution to ∆C = 2 is generated by CKM mixing, starting from the leading term in eq. (2.3). This can be written as Remarkably, for g /g q = O(1) this result is compatible with the existing bounds from CP violation in D-D mixing that require Λ uc > 3 × 10 3 TeV [38]. For R 0 = 0.15 this fixes | gq g | 5.4.

Bounds from LFU and LFV in τ decays, and neutrino physics
LFU in τ decays has been tested at the permil level. Assuming λ ij is negligible if i = e or j = e, and imposing |λ q ij | |V * 3i V 3j |, such tests can be used to set stringent limits on |λ µµ | and |λ τ µ |. Moreover, a strong limit on the product |λ τ µ ||λ µµ | follows from the upper bound on B(τ → 3µ).
The relevant modified effective Lagrangians are As far as LFU tests are concerned, the observable we consider is [39] .9726 is a phase space factor, and we summed over neutrinos of arbitrary flavor. The numerical result on the r.h.s. of eq. (3.14) is obtained using PDG data [5]. Expanding to first order in R 0 and assuming |λ τ µ | 2 |λ µµ | we obtain
As far as LFV is concerned, the B(τ → 3µ) < 2.1 × 10 −8 bound [5] implies Λ τ µ > 11 TeV, that can be translated into If |λ µµ | assumes the maximal allowed by eq. (3.15), we are left with the bound |λ τ µ | < ∼ 0.15. The latter is compatible with the hypothesis that λ ij admits an expansion similar to that of λ q ij in eq. (2.3), or that the leading breaking of the U(2) flavor symmetry is determined by spurions transforming as U(2) doublets.

New-physics effects in
The effective Lagrangian encoding NP effects in b → s + − ( = e, µ, τ ) is Using ∆L b→s + − to determine modified matching conditions for the Wilson coefficients of the most general b → s + − effective Hamiltonian, where

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Present b → sµ + µ − anomalies and R µ/e K seems to indicate a LF non-universal modification in the Wilson coefficients C µ 9 compared to the SM (see e.g. ref. [14,15]). However, a good fit to present data is also obtained assuming ∆C µ 9 = −∆C µ 10 = 0 and ∆C e 9,10 = 0, that is compatible with the modification expected in our NP framework for |λ µµ | |λ ee |. The best fit thus obtained implies ∆C µ 9 = −∆C µ 10 = −0.53 ± 0.18 [43,44]. In order to reproduce this result within our model we must impose This result is in some tension with the bounds on |λ µµ | and |λ q bs | dictated by LFU in τ decays and ∆m Bq mixing, respectively. To express this tension more clearly, it is convenient to normalize eq. (3.23) to the maximal value of |λ q bs | allowed by ∆m Bq mixing. This leads to λ q bs |λ q bs | max R 0 0.15 that should be compared with the constraint on |λ µµ | from eq. (3.15). Given the different scaling of eq. (3.24) and eq. (3.15) in terms of g /g q , the tension decreases for |g /g q | < 1.
As far as b → sτ + τ − decays are concerned, for R 0 = 0.15 and g q = g , we find Thus if λ q bs is close to |λ q bs | max , as favored by b → sµ + µ − anomalies and R µ/e K , depending on arg(λ q bs ) we have two very different non-standard predictions for b → sτ + τ − decays. In the case of maximal constructive interference of NP and SM amplitudes, b → sτ + τ − rates could be enhanced up to a factor ≈ 5 over the SM; in the case of maximal destructive interference, b → sτ + τ − rates could be strongly suppressed (even less than 1/10) compared to the SM expectation. This possible enhancement or suppression would hold also for the b → sν τντ rates, but it would appear "diluted" by a factor of ≈ 3 in the measurable b → sνν rates summed over all neutrino species.
In principle, the effective four-fermion Lagrangian in eq. (2.1) could allow also FCNC-LFV transitions of the type b → s ± i ∓ j , with the largest amplitude expected for b → sτ ± µ ∓ . The latter can be estimated by means of eq. (3.21), with the replacement λ µµ → λ τ µ . Given the constraint on |λ τ µ | in eq. (3.18), we find that FCNC-LFV helicity-conserving transitions (B → Kτ ± µ ∓ , B → K * τ ± µ ∓ , . . . ) can have rates which are at most 10% of those of the corresponding di-muon modes in the SM. Similarly, we find B(B s → τ ± µ ∓ ) < ∼ 10 −8 . These bounds makes the experimental search of these FCNC-LFV transitions very challenging, at least in the short term. We also note that such bounds are saturated only if B(τ → 3µ) is just below its current experimental bound.

Combined fit and discussion
The low-energy observables discussed above depend on the three flavor-non-universal couplings λ q bs , λ µµ , λ τ µ , and the two flavor-independent combinations  Table 1. Observables entering in the fit with their experimental bound (assuming the uncertainties follow the Gaussian distribution) and the expression in terms of the parameters of our model.
which we assume to be bounded by | ,q | < 2. We have performed a combined fit of these parameters using the experimental constraints reported in table 1. For simplicity, we have assumed Gaussian errors for all the observables. The preferred region of the model parameters (x α ) has been determined minimizing the χ 2 distribution The best-fit point is found for The χ 2 improvement of the best-fit point with respect to the SM limit is χ 2 (x SM ) − χ 2 (x BF ) = 18.6 for 5 d.o.f., which corresponds to a p-value for the SM hypothesis of 0.002. In figure 1 we show the 68%CL and 95%CL regions in the ( q , ), (λ q bs , λ µµ ), (λ µµ , λ τ µ ), and (∆C µ 9 , ∆R Bs ) planes, after having marginalised over the other parameters. The best-fit point implies a small non-standard contribution to C µ 9 . This is because of the bounds on |λ µµ | and |λ q bs | dictated by LFU in τ decays and ∆m Bq mixing (see section 3.4). However, in the 95%CL (68%CL) preferred region of the model parameters the effective coupling |λ µµ | can exceed 0.10 (0.05). In this case ∆C µ 9 can be within 1σ or 2σ of its central value (see right panels in figure 1). 7 In summary, we find that the effective Lagrangian in eq. (2.1) provides a significantly improved fit to low-energy data. It is worth stressing that, even from the EFT point of view, this model is highly constrained given the underlying set of dynamical hypotheses. As a result, the model leads to a series of predictions often different (more precise) than those obtained using more general EFT approaches (see e.g. [20,21,31]). The main predictions, 7 A "perfect fit" of ∆C µ 9 can be obtained extending the minimal version of the model, at the cost of introducing more free parameters. In particular, a natural extension is obtained with the inclusion of a SU(2)L singlet, coupled to the current J 0 µ obtained by J a µ in eq. (2.2) with the replacement T a → 1, g q → g q .  Figure 1. Results of the low-energy fit in table 1: 68%CL (green) and 95%CL (yellow) allowed regions in the ( q , ) plane (upper-left plot), (λ q bs , λ µµ ) plane (upper-right plot), (λ µµ , λ τ µ ) plane (lower-left plot), and in the (∆C µ 9 , ∆R Bs ) plane (lower-right plot), after having marginalised over the variables not shown. The black dots represent the best-fit points for these 2d likelihoods. In the upper-right plot, the solid, dashed, and dotted red lines represent the iso-lines respectively for the best-fit, 1-and 2-σ ranges for ∆C µ 9 , with fixed R 0 = 0.15. In the lower-right plot, the dashed and solid blue lines represent the 68%CL and 98%CL regions for ∆C µ 9 and ∆R Bs as favored by b → sµ + µ − and ∆m Bs data. which can be used to test the model in a more stringent way in view of future data, can be summarized as follows: FCNC. The modification of the b → s + − operators are purely left-handed. This implies, in particular, ∆C µ 9 = −∆C µ 10 , hence a suppression (in the 10%-20% range) of B(B s,d → µ + µ − ) rates compared to their SM expectations. The central value of the anomaly in R µ/e K is likely to decrease (to ∼ 10% or less). The NP contribution to the b → sτ + τ − amplitude is likely to be close, in magnitude, to the SM one. This implies a rate enhancement of at most a factor of ≈ 5 compared to the SM (constructive interference) or a strong suppression (destructive interference). The magnitude of the FCNC-LFV transitions b → sµ ± τ ∓ is at most 10% (in the rates) compared to the b → s + − ones.
Meson-antimeson mixing. A O(10%) deviation from the SM is expected B s mixing, if the anomaly in R µ/e K persists. According to the most plausible breaking structure of the U(2) q symmetry, this deviation should be present also in B d mixing and should preserve the relation ∆M Bs /∆M B d = (∆M Bs /∆M B d ) SM . The D-D mixing amplitude should acquire a CP-violating phase, whose magnitude could be just below the current experimental bounds.
τ decays. The τ → 3µ and µ → 3e processes are generated at the tree level (contrary to LFV dipole transitions i → j γ) and could be close to the present experimental bounds, although no precise correlations with other observables can be derived at present. If the anomaly in R µ/e K persists, violations of LFU in τ → µνν vs. τ → eνν are expected to be just below the current experimental bounds.

Bounds from LEP-I
Since the couplings of the heavy vector with SM fermions in this model are strongly non-universal, we cannot apply the LEP-I constraints as encoded in the bound on the S-parameter. Instead, we consider the non-universal fit of LEP-I data performed in the context of dimension-6 operators in ref. [46]. To do this, we translate the effective Lagrangian of eq. (2.10) to the Higgs basis used for the fit.
Since the constraints on the Z couplings to third generation (left-handed) quarks and leptons are of the same order as the bounds on the couplings to lighter fermions, while in our model the third generation is the one with biggest couplings, the strongest constraints on the model will arise from the bounds on Z couplings to third generation fermions. This motivates us to simplify the analysis of the EFT fit by neglecting λ q, ij for i, j = 3. In this limit the fit only depends on these two combinations of parameters: and the correlation is negligible. We introduced the adimensional parameters X ≡ g X m W gm V , with X = , q, H. With this notation the constraint in eq. constraints with the ones from flavor physics and show the 68%CL and 95%CL allowed regions in the ( q , H ) plane. From this we conclude that | H | 5 × 10 −3 . This result allows us to conclude that, in absence of new degrees of freedom in the model, the massive vectors decay dominantly to SM fermions by means of the last term (V a µ J a µ ) in eq. (2.9).

High-energy searches
We parametrize massive vector boson couplings to SM fermions (in their mass-eigenstate basis) as follows With this definition, the two body V →f i f j decay width is

4)
N C is the dimension of the color representation of the fermions, and we have assumed Due to the approximate U(2) q × U(2) symmetry, the total decay width of the vector bosons is dominated by decays to third-generation fermions. In the limit m 2 The neutral vector boson predominantly decays to τ + τ − ,ν τ ν τ ,bb, andtt final states. The relative impact of the leptonic and hadronic decay modes is driven by the ratio g /g q  (and the phase-space corrections to the large top-quark mass). The decay to a muon pair is parametrically suppressed by the smallness of λ µµ . In particular, the following relation holds The dominant charged vector decay modes are tb and τ + ν τ . In the following we assume m V + > m t , such that the V + state cannot be produced on-shell from top decays. We checked by explicit computation that when this criteria is satisfied the corrections to t → bτ + ν τ decay are well below present experimental sensitivity. The decays of both charged and neutral states to SM gauge bosons are strongly suppressed due to the strong limits on the H parameter from electroweak precision data (see section 4.1). The single vector bosons production at the LHC is dominated by Drell-Yan type processes, i.e. pp → V + X, where X stands for additional hadronic activity. While resonance searches in general impose severe limits on sequential (SM-like) W and Z bosons, we find significantly milder limits within our model. This is because of the specific flavor structure which suppresses both the production cross section and the decays into muon or electron final states.
In order to derive the present collider limits on the model, we have confronted the predictions of the model to a number of ATLAS and CMS searches for heavy W and Z resonances [47][48][49][50][51][52][53]. To this purpose, we have implemented the model in Universal FeynRules Output (UFO) [56] [57] package to simulate the tree-level pp → V ± and pp → V 0 production at √ s = 8 TeV in the 5-flavor scheme. Finally, we have validated the implementation of the heavy vector triplet couplings to fermions by simulating decays and comparing the numerical results with the analytic expressions in eq. (4.3).
In figure 3 (left) we show the predicted cross sections (in pb) for pp → V ± (pp → V + plus pp → V − ) and pp → V 0 , as obtained in the limit λ q ij = δ i3 δ 3j (i.e. exact flavor symmetry, but for the breaking terms induced by the SM Yukawa couplings). In this limit -14 -

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the production cross sections are completely determined by g q and m V . As can be seen, the neutral cross-section is about 100 times larger than the charged one. This is because the V 0 state is produced by bottom-bottom fusion, that is allowed in the limit of exact flavor symmetry, while the leading V ± production channel is bottom-charm fusion, that is suppressed by |V cb | at the amplitude level.
The search for W → tb with the ATLAS detector at 8 TeV and 20.3 fb −1 of data excludes a left-handed W boson with a mass of 500 (1000) GeV if σ(pp → W ) × B(W → tb) > 3.3 (0.19) pb [47]. In addition, the CMS search for W → τ ν performed at 8 TeV with an integrated luminosity of 19.7 fb −1 excludes a W boson with mass 300 (500) GeV if σ(pp → W ) × B(W → τ ν) > 4 (0.1) pb [48]. Comparing these limits with the predicted cross sections shown in figure 3 (left), we conclude that these searches have little impact on our model. We also checked that the ATLAS search for W → µν [49] has no relevance due to the limit on the |λ µµ | coupling.
The resonance searches for neutral vector bosons are more relevant due to the larger expected cross section. The ATLAS search for Z resonances decaying to τ + τ − using 19.5 − 20.3 fb −1 of 8 TeV data [50] sets an important constraint on the parameter space of the model. The exclusion limits (under the assumption of a narrow resonance) are shown in figure 4 in cyan solid (dashed) line assuming B(Z → τ + τ − ) = 0.01 (0.10). The region above these lines is excluded at 95%CL. The exclusion limits start from m V = 500 GeV because ref. [50] reports the limits on σ × B only above this mass. To overcome this problem, we urge the experimental collaborations to extend the search for Z resonances even in the low mass region. To extract the present limits for m V < 500 GeV, we use the CMS search for the neutral MSSM Higgs boson decaying to a pair of tau leptons [54] at 8 TeV and 19.7 fb −1 luminosity. The collaboration reports a model independent limit on the bb-induced production cross section times B(H → τ + τ − ) (assuming a narrow resonance) in the region 100 GeV < m H < 1 TeV. We have performed a parton-level MadGraph simulation to compare the kinematics of the τ + τ − final state produced by a scalar and a vector resonance (of mass 200 GeV). Having found small differences, we have re-interpreted the CMS bound into the σ × B limit for our model reported in figure 4. In particular, the region above pink solid (dashed) line is excluded assuming B(Z → τ + τ − ) = 0.01 (0.10). While the searches for dimuon resonances are usually more sensitive than the τ + τ − ones (see for instance [51]), we find them less relevant for our model due to smallness of λ µµ . On the other hand, dijet [52] and tt [53] resonance searches set limits on the cross section times branching ratio for m V ∼ 1 TeV of the order of 1 pb.
The impact of the direct searches in the τ + τ − channel on the parameter region preferred by flavor data is illustrated in figure 4. The wide light green region is obtained imposing R 0 = 0.14 ± 0.04 (68% CL region) and g q , g < √ 4π. The narrower dark green (yellow) band is the region for which R 0 is within 68% CL (95% CL) and g q = g . In the minimal model, the predicted V 0 → τ + τ − branching ratio for g q = g is B(V 0 → τ + τ − ) ≈ 1/8. Comparing with the exclusion curve obtained for B(Z → τ + τ − ) = 0.1 we deduce that the minimal model is ruled-out for g q = g . The situation improves for g g q (a configuration also preferred by flavor data, albeit with the lower bound g g q /5.4 from D-D mixing, see section 3.2), given B(V 0 → τ + τ − ) ≈ (1/8) × (g /g q ) 2 . However, it is not possible to  completely evade the bounds for perturbative values of the couplings (g q , g < √ 4π) in the region preferred by flavor data.
There are different ways to evade the LHC limits on pp → V 0 → τ + τ − going beyond the minimal model. The simplest possibility is to add new V 0 decay channels, say to a dark sector. This would result into lower values of B(V 0 → τ + τ − ). As can be seen in figure 4 if, for g q = g , B(V 0 → τ + τ − ) decreases to 0.01, then there are regions of the parameter space that are allowed, both at low and at high masses. Another option is to consider a heavy V 0 in the limit of a strongly coupled theory (Γ V ∼ M V ). In this case the resonance becomes broad and the limits obtained assuming a narrow state no longer holds. A third possibility would be to add an additional neutral heavy vector, e.g. a SU(2) L singlet, close in mass to the neutral component of the triplet, with couplings tuned to interfere destructively with V 0 in the pp → τ + τ − + X cross section.
We finally comment about the pp → V V (pair production) process at the LHC. This proceeds via: i) t-channel quark-exchange diagrams, controlled by eq. (4.2), and ii) schannel diagrams with off-shell SM electroweak gauge bosons. In the limit of no mixing with EW gauge bosons, g H ≈ 0, and neglecting contributions from additional non-minimal operators [45], the relevant interactions of the heavy vectors with the SM electroweak gauge bosons described by eq. (2.9) are where g is the SU(2) L coupling constant and s θ (c θ ) is sine (cosine) of the Weinberg angle.
As for the single V production, we have used MadGraph to simulate pp → V + V − and pp → V ± V 0 (that is, pp → V + V 0 plus pp → V − V 0 ). In figure 3 (right) we show the predicted cross sections (in pb) for 8 TeV proton-proton collisions, as obtained in the limit λ q ij = δ i3 δ 3j . As illustrated with the vertical iso-lines, the cross sections are dominated by -16 -

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s-channel diagrams for small g q couplings. On the other hand, for large couplings there is a substantial contribution from the bb induced t-channel diagram to V + V − production. Based on the predicted cross sections, we conclude that the single production is more relevant compared to the pair production for the interesting region of the parameter space.

Conclusions
Lepton Flavor Universality is not a fundamental symmetry: within the Standard Model it is an approximate accidental symmetry broken only by the Yukawa interactions. This specific symmetry and symmetry-breaking pattern results in tiny deviations from LFU in helicity-conserving amplitudes, within the SM, and it implies that LFU tests are clean probes of physics beyond the SM. Motivated by a series of recent experimental results in B physics pointing to possible violations of LFU, both in charged and in neutral currents, in this paper we have consider a simplified dynamical model able to describe these effects in a unified way. In particular, we have shown that a SU(2) L triplet of massive vector bosons, coupled predominantly to third generation fermions (both quarks and leptons), can significantly improve the description of present data.
The proposed model has a series of virtues compared to previous attempts to describe such effects in terms of New Physics: i) it connects the breaking of LFU between charged and neutral currents, and between semi-leptonic and purely leptonic processes; ii) it is based on a simple flavor symmetry, whose breaking terms are related to the structure of the SM Yukawa couplings, both in the quark and in the lepton sector; iii) it connects low-energy deviations from the SM to direct searches for NP at high p T . The constrained structure of the model makes it highly non trivial to satisfy all existing bounds and, at the same time, accommodate deviations from the SM as large as indicated by the central values in eqs. (1.1)-(1.3). We find that this happens quite naturally in the case of charged currents, both at low and at high energies. The situation is more problematic in the case of neutral currents. On the one hand, the maximal deviations from unity in R µ/e K barely exceed 10%. On the other hand, the minimal version of the model is ruled out by the direct searches for resonances decaying into τ + τ − at ATLAS and CMS. As discussed, both these issues can be improved with less minimal versions of the model, at the cost of introducing more free parameters.
One of the most remarkable aspects of the minimal version of the model is the welldefined pattern of deviations in low-energy processes listed at the end of section 3.5. This pattern is largely insensitive to possible extensions of the model necessary to overcome the constraints from direct searches. It mainly reflects the symmetry structure of the model and could be used, in the near future, to verify or falsify this framework with more precise data.
Besides the specific predictions of the proposed model, we stress the importance of future experimental tests of LFU at low-energies, and dedicated searches for flavor-nonuniversal phenomena at high energies. On the low-energy side, we urge the experimental collaborations to re-analyze charged-current B decays without assuming lepton flavor universality. On the high-energy side, we encourage the search for deviations from the SM -17 -

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in τ + τ − and tt invariant-mass distributions, relaxing the hypothesis of narrow resonances and covering also the region of low invariant masses.