Phenomenology of an $SU(2) \times SU(2) \times U(1)$ model with lepton-flavour non-universality

We investigate a gauge extension of the Standard Model in light of the observed hints of lepton universality violation in $b \to c \ell \nu$ and $b \to s \ell^+ \ell^-$ decays at BaBar, Belle and LHCb. The model consists of an extended gauge group $\mathrm{SU(2)}_{1} \times \mathrm{SU(2)}_{2} \times \mathrm{U(1)}_Y$ which breaks spontaneously around the TeV scale to the electroweak gauge group. Fermion mixing effects with vector-like fermions give rise to potentially large new physics contributions in flavour transitions mediated by $W^{\prime}$ and $Z^{\prime}$ bosons. This model can ease tensions in $B$-physics data while satisfying stringent bounds from flavour physics, tau decays, and electroweak precision data. Possible ways to test the proposed new physics scenario with upcoming experimental measurements are discussed. Among other predictions, the lepton flavour violating ratios $R_M$, with $M = K^*, \phi$, are found to be reduced with respect to the Standard Model expectation $R_M \simeq 1$.


Introduction
The Standard Model (SM) of particle physics, based on the SU(3) C × SU(2) L × U(1) Y gauge group, is an extremely successful theory that accounts for a wide range of high energy experiments at both the intensity and energy frontiers. It is nevertheless a theory that is widely considered to be incomplete, and manifestations of new physics (NP) are expected to show up around the TeV scale.
A large class of particularly attractive NP theories consider extensions of the SM where its gauge group is embedded into a larger one which breaks to the SM (directly or via various steps) at or above the TeV scale. In this view, the SM is seen as an effective model valid at low energies. These constructions include Grand Unified Theories (GUT), composite models and string-inspired models. Interestingly, when the last breaking of the extended gauge group occurs around the TeV scale, a plethora of observables are generally predicted. In particular, flavour physics observables constitute a powerful probe to test these models due to the impressive precision and reach of current experiments.
In this article we present a detailed phenomenological analysis focused on flavour observables of a minimal extension of the SM electroweak gauge group to SU(2) 1 × SU(2) 2 × U(1) Y . We remain agnostic as to the origin of such a gauge group but assume it is broken around the TeV scale. Models based on an extra SU(2) factor have been considered since a long time and constitute some of the most studied NP theories as they are predicted by various wellmotivated frameworks, such as SO (10) or E 6 GUTs. Depending on how the SU(2) and U (1) factors are identified, we can have for instance Left-Right [1] and Un-unified [2] schemes (for a general classification, cf. Ref. [3]). The extra SU(2) factor implies the existence of new force carriers in the form of heavy partners of the SM W and Z bosons. In general, their couplings to matter are dictated by the choice of representations of the SM fields and the exotic new fields (if any). In any case, a rich phenomenology is predicted.
The model we will analyse was first presented in Ref. [4]. While the construction of the model has been motivated mainly by recent anomalies in B decays, we will carry out here a generic analysis of the model and impose the constraints arising from these hints only as a secondary step.
The salient features of our model are summarised as follows: • The extended gauge symmetry SU(2) 1 × SU(2) 2 × U(1) Y spontaneously breaks at the TeV scale to the SM electroweak group following the pattern • The SM fields are all charged under one of the SU(2)'s only, with the same quantum numbers they have in the SM, whereas newly introduced vector-like fermions are charged similarly to the lepton and quark doublets but under the other SU(2) group.
• Fermion mixing effects (facilitated by the same scalar field which breaks the original group) between the exotic and SM fermions act as a source of flavour non-universal vector currents by modulating the couplings of the SM fermions to the new gauge bosons.
Unified explanations of both sets of anomalies are much more scarce. This is due to the difficulty of accounting for deviations of similar size in processes that take place in the SM at different orders: loop level for R K and tree-level for R(D ( * ) ). Nevertheless, among the proposed models we find those based on leptoquarks [72][73][74][75][76][77][78], an extended perturbative gauge group [4], or strongly-interacting models [79]. An effective field theory approach has been adopted in Refs. [72,[80][81][82] and some observations about the relevance of quantum effects have been given in Ref. [83].
In our model, the massive gauge vector bosons arising from the breaking of the extended gauge group mediate flavour transitions at tree-level as shown in Figure 1, providing a possible explanation to the deviations from the SM observed in B-meson decays [4].
The plan of the paper is as follows: in Section 2 we present the model in detail. We derive the gauge boson and fermion masses and mixings, as well as the required textures in Section 3. A detailed description of the flavour and electroweak observables included in the global fit is given in Section 4. Our global fit main results and predictions are presented in Section 5 and Section 6, respectively. Finally, in Section 7 we provide our conclusions. Details of the model are provided in the Appendices.

Description of the model
We consider a theory with the electroweak gauge group promoted to SU(2) 1 ×SU(2) 2 ×U(1) Y . The factor U(1) Y corresponds to the usual hypercharge while the SM SU(2) L is contained in the SU(2) product. The gauge bosons and gauge couplings of the extended electroweak group will be denoted as: where i = 1, 2, 3 is the SU(2) index. All of the SM left-handed fermions transform exclusively under the second SU(2) factor, i.e.
where the representations refer to SU(3) C , SU(2) 1 and SU(2) 2 , respectively, while the subscript denotes the hypercharge. The SM doublets q L and L can be decomposed in SU(2) 2 components in the usual way, In addition, we introduce n VL generations of vector-like fermions transforming as For the moment we take the number of generations n VL as a free parameter to be constrained by phenomenological requirements. Symmetry breaking is achieved via the following set of scalars: a self-dual bidoublet Φ (i.e., Φ = σ 2 Φ * σ 2 , with σ 2 the usual Pauli matrix) and two doublets φ and φ , which we decompose as: withΦ 0 = (Φ 0 ) * and Φ − = (Φ + ) * . We summarise the particle content of the model in Table 1.

Yukawa interactions
The SM fermions couple to the SM Higgs-like φ doublet with the usual Yukawa terms, withφ ≡ iσ 2 φ * . The y u,d,e Yukawa couplings represent 3 × 3 matrices in family space. The vector-like fermions, on the other hand, have gauge-invariant Dirac mass terms, and our choice of representations allows us to Yukawa-couple them to the SM fermions via and where λ q, and y u,d,e are 3 × n VL and n VL × 3 Yukawa matrices, respectively. After spontaneous symmetry breaking, these couplings will induce mixings between the vector-like and SM chiral fermions. This is crucial for the phenomenology of the model, in particular in its flavour sector, as will be clear in the next sections.

Scalar potential and symmetry breaking
The scalar potential can be cast as follows: We will assume that the parameters in the scalar potential are such that the scalar fields develop vevs in the following directions: Assuming u v φ , v φ , the symmetry breaking proceeds via the following pattern: with the assumed vev hierarchy u ∼ TeV v 246 GeV. With this breaking chain, the charge of the unbroken U(1) em group is defined as with T a 3 the diagonal generator of SU(2) a . In the first step, the original SU(2) 1 × SU(2) 2 group gets broken down to the diagonal SU(2) L . Under the diagonal sub-group, φ and φ transform as doublets and, as usual with two-Higgs doublet models (2HDM), we parametrize their vevs as where Since the two doublets transformed originally in a 'mirror' way under the two original SU(2) factors, it is clear that the ratio between their vevs, tan β = v φ /v φ , controls the size of the gauge mixing effects. In particular, the limit tan β = g 1 /g 2 corresponds to the purely diagonal limit with no gauge mixing, see Subsection 3.2 for more details.
The scalar fields {φ, Φ, φ } contain 12 real degrees of freedom, six of these become the longitudinal polarization components of the W ( )± and Z ( ) bosons. In the CP-conserving limit the scalar spectrum is composed of three CP-even Higgs bosons, one CP-odd Higgs and one charged scalar, forming an effective (constrained) 2HDM plus CP-even singlet system. The scalar sector will present a decoupling behaviour, with a SM-like Higgs boson at the weak scale (to be associated with the 125 GeV boson) and the rest of the scalars at the scale u ∼ TeV. 1 Further details of the scalar sector are given in Appendix A.

Gauge boson and fermion masses and interactions
We now proceed to the analysis of the model presented in the previous section. Here we will derive the masses and mixing of the gauge bosons and fermions of the model, as well as the neutral and charged vectorial currents.

Fermion masses
We can combine the SM and the vector-like fermions as where i = 1, 2, 3, k = 1, . . . , n VL and I = 1, . . . , 3 + n VL . With this notation the fermion mass Lagrangian after symmetry breaking is given by The mass matrices are given in terms of the Yukawa couplings, vector-like Dirac masses and vevs as Note that we did not include any mechanism to generate neutrino masses, and consequently M N leads to three massless neutrinos and n VL heavy neutral Dirac fermions. It is nevertheless straightforward to account for neutrino masses without impacting our analysis and conclusions by including one of the usual mechanisms, such as the standard seesaw. In order to have a manageable parameter space and simplify the analysis we will assume that the Yukawa couplings of φ can be neglected, y u,d,e 0. This can be justified by introducing a softly-broken discrete Z 2 symmetry under which φ is odd and all the other fields are even. We take the Dirac masses of the vector-like fermions to be generically around the symmetry breaking scale u ∼ TeV.
The fermion mass matrices can be block-diagonalized perturbatively in the small ratio = v/u 1 by means of the following field transformations defined in terms of the unitary matrices Here the freedom in the definition of V 11 Q,L is removed by choosing it to be hermitian. Furthermore, u M Q,L is the physical vector-like mass at leading order in , and the matrices H f V and H f W are given by with F = Q, L and f = u, d, e. After the block-diagonalization, a further diagonalization of the SM fermion block can be done by means of the 3 × 3 unitary transformations As in the SM, only one combination of these transformations appears in the gauge couplings: the CKM matrix, V CKM = S u S † d .

Vector boson masses and gauge mixing Neutral gauge bosons
The neutral gauge bosons mass matrix in the basis V 0 = (W 1 3 , W 2 3 , B) is given by: This matrix has one vanishing eigenvalue, corresponding to the photon and two massive eigenstates which are identified with the Z and Z bosons. Before fully diagonalizing this mass matrix we consider first the rotation from (W 1 3 , W 2 3 ) to (Z h , W 3 ), with W 3 the electrically neutral SU(2) L gauge boson. In order to do this we have to study the first symmetry breaking step, i.e. u = 0 and v = 0, diagonalize the top-left 2 × 2 block and identify the massless state with W 3 (the SU(2) L group remains unbroken in the first step). As a result we get: with n 1 = g 2 1 + g 2 2 and the gauge coupling of SU(2) L taking the value g = g 1 g 2 /n 1 . In the (Z h , W 3 , B) basis, the rotation from (W 3 , B) to (Z l , A) is just like in the SM and we obtain: where n 2 = g 2 + g 2 and the weak angle is defined as usual:ŝ W = g /n 2 andĉ W = g/n 2 .
We are now in condition to write the neutral gauge boson mass matrix in the (Z h , Z l , A) basis where it takes the form: We see from this mass matrix that in the particular limit Moreover, we can extract the Z l − Z h mixing. The mass eigenvectors (Z , Z) are given, in terms of (Z h , Z l ), by: with the mixing suppressed by the ratio ≡ v/u, We define the parameter controlling the mixing as In the limit ζ → 0, the SU(2) L sub-group corresponds to the diagonal subgroup of the original SU(2) product and gauge mixing vanishes. As anticipated in Section 2, ζ → 0 corresponds to the limit tan β → g 1 /g 2 . Finally, the masses of the neutral massive vector bosons are given by

Charged gauge bosons
In the basis , the charged gauge boson mass matrix is given by As before, it is convenient to work in the basis (W h , W l ) where the SU(2) L gauge boson appears explicitly. To obtain this basis in terms of the original one, we set v = 0, diagonalize the mass matrix and associate the null eigenvalue to W l (SU(2) L remains unbroken in the first stage of symmetry breaking). We get: In the basis (W h , W l ) the mass matrix reads: The W l − W h mixing presents the same structure as in the neutral gauge boson sector and reads: such that the physical eigenstates are given by: with masses

Neutral currents
The neutral currents of the fermions are given by with ψ = U, D, E, N , and e = gg /n 2 and Q ψ denoting the electric coupling and the electric charge of the fermions, respectively. Applying the transformations in Eqs. (19) and (25) we can easily translate the above interactions to the fermion mass eigenbasis Here m f denotes the mass of a SM fermion with f = u, d, e, and we introduced the following definitions: with F = Q, L, and finallyĝ ≡ gg 2 /g 1 .

Charged currents
Similarly, the charged currents take the following form and, in the fermion mass eigenbasis (see Eqs. (19) and (25)), we have

Flavour textures for the gauge interactions
In order to accommodate the hints of lepton universality violation from the recent anomalies in B decays without being in tension with other bounds, we require negligible couplings of the new gauge bosons to the first family of SM-like leptons and a large universality violation among the other two. We now derive the conditions on the number of generations of the exotic fermions to accommodate such constraints. Using Eqs. (42) and (20), the matrix ∆ q, , that parametrize NP contributions to the lefthanded gauge interactions with SM fermions, can be readily written in the following form where the second term is the source of lepton non-universality induced by the mixings between the SM and vector-like fermions generated by the λ q, Yukawa couplings. On the other hand, right-handed couplings involving SM fermions, controlled by O f f R , are mass suppressed and they can be neglected for the interactions we are considering.
If we consider the minimal scenario with n VL = 1, the Yukawa couplings λ q, can be written generically as Here ∆ d,e , ∆ s,b and ∆ µ,τ are free real parameters, and without loss of generality we have chosen an appropriate normalization factor to simplify the expression of ∆ q, . We have also ignored possible complex phases in the couplings since we are not interested in CP violating observables. From Eq. (47) it is then clear that, for n VL = 1, NP contributions to the lefthanded gauge couplings to SM fermions are given by As we can see, in the limit of no gauge boson mixing, NP contributions to the first family of SM fermions can only be suppressed if we fix ∆ d,e 1 and ∆ s,µ , ∆ b,τ 1 which then implies approximate universal couplings for the second and the third families. Hence, we need at least two generations of vector-like fermions in order to have enough freedom to accommodate the observed hints of lepton universality violation.
In the rest of this article we will take the minimal setup consisting of n VL = 2 since there is no compelling reason to assume additional vector-like generations. Moreover, in order to reduce the number of free parameters in the analysis we choose the following texture for the Yukawa matrices λ q, : where, again, ∆ s,b and ∆ µ,τ are free real parameters and the normalization factor is chosen for convenience. 2 The left-handed currents, parametrized in terms of O Q,L L (see Eq. (42)) now read which, by construction, provide the desired patterns for the NP contributions to accommodate the data.

Flavour constraints
We consider in our analysis flavour observables receiving new physics contributions at treelevel from the exchange of the massive vector bosons. Additionally, we consider bounds from electroweak precision measurements at the Z and W pole which are affected in our model due to gauge mixing effects. Regarding electroweak precision observables at the Z and W pole, we use the fit to Zand W -pole observables performed in Ref. [84]. The fit includes the observables listed in Tables 1  and 2 of [84], and provides mean values, standard deviations and the correlation matrix for the following parameters: the correction to the W mass (δm), anomalous W and Z couplings to leptons (δg W i L , δg Z i L,R ) and anomalous Z couplings to quarks (δg Zu i L,R , δg Zd i L,R ). The results for these "pseudo-observables" can be found in Eqs. (4.5-4.8) and Appendix B of Ref. [84]. The relevant expressions for these pseudo-observables within our model are given in Appendix B.
We collect the list of flavour observables included in our analysis in Table 2 and describe them in more detail in the following subsections.

Leptonic Tau decays
Leptonic tau decays pose very stringent constraints on lepton flavour universality [85]. We consider the two decay rates Γ(τ → {e, µ}νν), normalized to the muon decay rate to cancel the dependence on G F . We take the individual experimental branching ratios and lifetimes from the PDG [86]. For the branching ratios we take the result of the constrained fit, which gives a correlation of 14% between both measurements. Once normalized to the τ lifetime, the decay rates have a correlation of 45%, while the normalization to the muon decay rate has a minor impact on the correlation of the ratios because its uncertainty is negligible. The experimental results are summarized in Table 2.
In our model, we have: are given by The resulting predictions in the SM can be found in Table 2. Leading radiative corrections and W -boson propagator effects are included in the SM predictions [87][88][89][90].

d → u transitions
We consider the decay rates Γ(π → µν) and Γ(τ → πν), normalized to Γ(π → eν) in order to cancel the dependence on the combination G F |V ud |f π . These ratios constitute important constraints on flavour non-universality in d → u ν transitions. We calculate the experimental values for these ratios taking the averages for branching fractions and lifetimes from the PDG [86], and imposing the constraint B(π → eν) + B(π → µν) = 1. We find a correlation of 49% between both ratios. The corresponding results are summarized in Table 2.
The model predictions for these ratios are: where the Wilson coefficients C u i d j ab are given by For the SM contributions we follow Ref. [85]. We have: The calculation of δR π→e/µ relies on Chiral Perturbation Theory to order O(e 2 p 2 ) [91]. The radiative correction factor δR τ /π can be found in Ref. [92]. The SM predictions for both ratios are collected in Table 2.

s → u transitions
We consider the decay rates Γ(K → µν) and Γ(τ → Kν), normalized to Γ(K → eν) in order to cancel the dependence on the combination G F |V us |f K , as well as the semileptonic (K 3 ) ratio Γ(K + → π 0 µ + ν)/Γ(K + → π 0 e + ν). These ratios pose also important constraints on flavour non-universality. We take the experimental values for the decay rates Γ(K + → µ + ν), Γ(K + → π 0 e + ν) and Γ(K + → π 0 µ + ν) from the constrained fit to K + decay data done by the PDG [86], including the correlation matrix. The correlation between Γ(K + → µ + ν) and Γ(K + → e + ν) is calculated comparing the averages for the individual rates with the ratio given by the PDG, resulting in a correlation of 60%. Assuming no correlation between Γ(K + → e + ν) and the semileptonic modes, and assuming that the τ mode is uncorrelated to the K modes, we construct a 5 × 5 correlation matrix and calculate the three ratios of interest, including their 3 × 3 correlation matrix. These results are collected in Table 2.
The model predictions for these ratios are: with the Wilson coefficients C us ij given in Eq. (59). The SM contributions for the first two ratios are given by the analogous expressions to Eqs. (60,61) [91,92]. The SM contributions to K 3 are given by [93,94] where quantities I (2) encoding phase-space factors, electromagnetic and isospin corrections can be found in Refs. [93][94][95]. The numerical results for the SM contributions are collected in Table 2.
For D → K ν, we consider charged and neutral modes separately. For D + →K 0 + ν we take the separate branching ratios from the PDG assuming no correlation. For D 0 → K − + ν we take the results from the PDG constrained fit, including the 5% correlation. We construct the D + and D 0 ratios separately, obtaining Γ(D + →K 0 µ + ν)/Γ(D + →K 0 e + ν) = 1.05 (9) and (4). These two ratios, corresponding to the same theoretical quantity (isospin-breaking effects are neglected here), are combined according to the PDG averaging prescription. Since there is a ∼ 1 σ tension between both results, we rescale the error by the factor S = 1.3.
For D s → ν we take the individual branching fractions from the PDG, assuming no correlation. The resulting experimental numbers for both ratios are collected in Table 2.
The model predictions for these ratios are: with the Wilson coefficients C cs ij given in Eq. (59). Our SM prediction for the leptonic decay modes includes electromagnetic corrections following [96]. For the SM prediction of the semileptonic modes we use the BESIII determination of the form factor parameters in the simple pole scheme as quoted in HFAG [5]. The resulting SM predictions are given in Table 2.

b → s transitions
We consider here b → s transitions that are loop-mediated in the SM but receive NP contributions at tree-level in our model (via Z and Z with anomalous couplings). To the level of precision we are working, the normalization factors in the SM amplitude (G F and CKM elements) can be taken from tree-level determinations within the SM, and it is not necessary in this case to consider only ratios where these cancel out.

Mass difference in the B s system
The observable ∆M s constitutes a strong constraint on the Z sb coupling, independent of the coupling to leptons. In order to minimize the uncertainty from hadronic matrix elements, we consider the ratio ∆M s /∆M d . We note that within our model set-up, ∆M d does not receive NP contributions at tree-level.
The experimental value for the ratio is obtained from the individual measurements for ∆M d,s , which are known to subpercent precision [5]. The result is given in Table 2.
The theory prediction is given by: where the Wilson coefficients Here S 0 (x t ) = 2.322 ± 0.018 is the loop function in the SM [97]. The parameter ξ 2 = f 2 Bs B (1) B d is a ratio of decay constants and matrix elements determined from lattice QCD. We consider the latest determination of the parameter ξ from the FNAL/MILC collaborations [98]: ξ = 1.206(18) (6). The SM prediction is given by the first term in Eq. (68) and results in (∆M s /∆M d ) SM = 31.2(1.8).
Definitions, theoretical expressions and discussions on theoretical uncertainties can be found in Refs. [16,107]. We follow the approach of Ref. [19] for B → V form factors, and take into account the lifetime effect for B s measurements at hadronic machines [115] for B s → µµ [116] and B s → φµµ [117] decays.
We implement the fit in two different ways. First, we construct the full χ 2 as a function of the model parameters, including all theoretical and experimental correlations, exactly as in Ref. [16]. 3 Second, in order to provide simplified expressions to allow the reader to repeat the fit without too much work, we perform a global fit to the relevant coefficients of the effective weak Hamiltonian with We consider those coefficients receiving non-negligible NP contributions within our model, i.e. (C 9µ , C 10µ , C 9e , C 10e ), and provide the best fit points, standard deviations and correlation matrix. 4 These are collected in Table 2. The NP contributions to the Wilson coefficients Using these four coefficients as "pseudo observables" and constructing the χ 2 function leads to a linearised approximation to the fit. We have checked that the result of such a fit is in reasonable agreement with the full fit.
The experimental value for the inclusive ratio R(X c ) is obtained from the PDG averages for Br(b → Xτ + ν) and Br(b → Xe + ν). The allowed size of lepton flavour universality violating effects in b → c ν ( = e, µ) transitions is not trivial to account for given that experimental analyses tend to present combined results for the electron and muon data samples. This aspect was also stressed in Ref. [81]. Experimental results are however reported separately for the e and µ samples in an analysis performed by the BaBar collaboration [118]. We use the values of Br(B → D ( * ) ν) reported in Table IV of Ref. [118] to extract the ratios Γ(B → D ( * ) µν)/Γ(B → D ( * ) eν). The correlation between the two ratios is estimated from the information provided in [118], adding the covariance for the systematic and statistical errors. For the experimental values of R(D) and R(D * ) we consider the latest HFAG average [5]. The latter includes R(D) and R(D * ) measurements performed by BaBar and Belle [119,120], the LHCb measurement of R(D * ) [121], and the independent Belle measurement of R(D * ) using a semileptonic tagging method [122]. 5 The results are summarized in Table 2.
The model expressions for these ratios are: where the Wilson coefficients C cb ij are given in Eq. (59). We use the SM predictions of R(D) and R(D * ) obtained in Refs. [124,125]. Note that recent determinations of R(D) in Lattice QCD are compatible with the one used here [126,127]. For R(X c ) we use the SM prediction reported in Ref. [128]. For the ratios Γ(B − → D ( * ) µν)/Γ(B − → D ( * ) eν) we derive the SM predictions using the Caprini-Lellouch-Neubert parametrization of the form factors [129], with the relevant parameters taken from HFAG [5]. The resulting SM predictions are given in Table 2.

Lepton Flavour Violation
We consider current limits on the lepton flavour violating decays τ → 3µ and Z → τ µ. The decay Z → τ µ occurs due to gauge mixing effects. The decay rate for Z → τ µ ≡ We use the limit Br(Z → τ µ) < 1.2 × 10 −5 [86]. The decay τ → 3µ receives tree-level contributions from Z ( ) exchange, the decay rate is given by 5 New results for R(D * ) and the tau polarization asymmetry in B → D * τ ν decays (P τ ) using a hadronic tag have been presented by the Belle collaboration in Ref. [123]. The reported measurements are R(D * ) = 0.276 ± 0.034 +0.029 −0.026 and P τ = −0.44 ± 0.47 +0.20 −0.17 [123]. These measurements are not included in our analysis but would have a negligible impact if added given that the weighted average for R(D * ) remains basically the same and the experimental uncertainty in P τ is still very large. Note that the measured tau polarization asymmetry is well compatible with the SM prediction P τ = −0.502 +0.006 −0.005 ± 0.017 [25].

Fitting procedure
We first fix the values of g, g and the electroweak vev v with the values of {G F , α, M Z } reported in Table 3. The SU(2) 1 gauge coupling g 1 is then determined as a function of g 2 . The observables considered will depend on seven model parameters: The observables will also depend on the CKM inputs {λ, A,ρ,η}. We construct a global χ 2 function that includes information from electroweak precision data at the Z and W poles together with flavour data. It reads with Σ being the covariance matrix, O denoting the observables included in the analysis and O exp the corresponding experimental mean values. These are described in Section 4. The CKM inputs {λ, A,ρ,η} are included as pseudo-observables in the fit taking into account the values in Table 3. 6 The latter are reported in the formx +σ + −σ − . In the χ 2 we introduce the asymmetric error: σ ± = σ + (for x >x) and σ ± = σ − (for x <x).
The global fit takes into account then seven model parameters

Results of the fit
We restrict the parameter space to 500 GeV ≤ M Z ≤ 3000 GeV, g < g 2 < √ 4π, |∆ a | ≤ 3 and 0 ≤ ζ ≤ 1. The minimum of the χ 2 is found to be at with the CKM values {λ, A,ρ,η} within the 1σ range in Table 3. It is enlightening to characterise the best-fit point in terms of the couplings appearing in the Lagrangian. We find that the corresponding Yukawas are, up to a global sign, At the best-fit point we obtain χ 2 min = 54.8, to be compared with the corresponding value in the SM-limit χ 2 SM = 93.7. We derive contours of ∆χ 2 ≡ χ 2 −χ 2 min in two-dimensional planes after profiling over all the other parameters, taking ∆χ 2 = 2.3 for 68% confidence level (CL) and ∆χ 2 = 6.18 for 95% CL. Allowed regions for the model parameters obtained in this way are shown in Figure 2.
There is a four-fold degeneracy of the χ 2 minimum with the sign of ∆ µ,τ as no observable in the fit is sensitive to the relative sign between ∆ µ and ∆ τ . The allowed values of ∆ µ,τ lie in the region |∆ µ,τ | 1. While ∆ b is bounded to be very small ∼ 10 −2 , the allowed values for ∆ s are around −1. The negative sign obtained for the combination ∆ s ∆ b is related to the preference for negative values of C NP 9µ by b → s + − data. The allowed regions for the Wilson coefficients of b → s + − transitions from the global fit are shown in Figure 3. Note that with the assumed flavour structure we have the correlation C NP 10e = (4s 2 W − 1)C NP 9e . The relation C NP 9µ = −C NP 10µ on the other hand holds in our model only in the absence of gauge mixing effects. Departures from this correlation are possible as gauge mixing effects can be sizeable, see Figure 3 (left).
Allowed values at 68% and 95% CL for R K and R(D * ) are shown in Figure 4. The best fit point presents a sizeable deviation from the SM in R K in the direction of the LHCb measurement while the ratios R(D * ) are SM-like. Note that the NP scaling of R(D) is the same as for R(D * ) because the W couplings are mostly left-handed, with the right-handed couplings suppressed by m 2 f /u 2 . A significant enhancement of R(D ( * ) ) is possible within the allowed parameter region. The model presents a positive correlation between R K and  R(D ( * ) ) so that R K is above its best-fit value whenever R(D ( * ) ) gets enhanced. The ratios Γ(B → D ( * ) µν)/Γ(B → D ( * ) eν) are found to be SM-like with possible deviations only at the ∼ 1% level. As expected, R(X c ) and R(D ( * ) ) show a strong correlation, in the region of the parameter space where R(D ( * ) ) accommodates the current experimental values one obtains a slight tension in R(X c ) with experiment. The flavour observables with light-mesons and leptonic τ -decays are found to be in good agreement with the SM and experiment, we show the resulting allowed values for K → µν/K → eν and τ → µνν/µ → eνν as an example in Figure 4. As noted in Ref. [4], gauge mixing effects play a crucial role in the possible enhancement of R(D ( * ) ) in this model. In Figure 4 we also show the results of the global fit for R(D * ) as a function of the parameter controlling the size of gauge mixing effects ζ. Having an enhancement of R(D * ) of order ∼ 20% as suggested by the experimental measurements is only possible for ζ 1. The situation is very different for R K , with the parameter ζ playing no major role in this case as shown in Figure 4. We find that the allowed points from the global fit accommodating both R K and R(D ( * ) ) within 2σ lie within a very restricted region: The Z mass and the SU(2) 2 gauge coupling g 2 are positively correlated, going from g 2 ∼ 1 for M Z ∼ 500 GeV up to the perturbativity limit g 2 ≤ √ 4π for M Z ∼ 1700 GeV. A limit on tan β can be derived in this region using Eq. (32), we get tan β ∈ [0.2, 0.65]. Similarly, in this region the SU(2) 1 gauge coupling satisfies 0.66 ≤ g 1 ≤ 0.78 and the combinationĝ = gg 2 /g 1 is found to be within 1 ≤ĝ ≤ 3.4. Note that the Z and W interactions with the SM fermions are proportional to 1 − ∆ 2 a , see Eq. (51). In the parameter space region where both R K and R(D ( * ) ) are accommodated within 2σ, the massive gauge bosons, Z and W , couple predominantly to the third fermion generation.

Predictions
In the following we take the current measured values of R K and R(D ( * ) ) at face value, focusing on the parameter space region described in Eq. (82). We are interested in possible signatures that can be used to test or falsify this scenario with upcoming measurements at the LHC and flavour factories. This gives rise to a clean prediction which is compatible with current data [5]. The inclusive ratio R(X c ) can provide an additional handle to test the proposed scenario. The model gives rise to an enhancement in R(X c ) within the parameter space region considered, we obtain 0.24 ≤ R(X c ) ≤ 0.29. The Dirac structure of the new physics contributions can also be tested by using information from the q 2 ≡ (p B − p D ( * ) ) 2 spectra and by measuring additional observables that exploit the rich kinematics and spin of the final state particles. The differential decay rate for B → D ( * ) τ ν is affected in the model with a global rescaling factor, implying that forward-backward asymmetries as well as the τ and D * polarization fractions are expected to be as in the SM. For recent studies of differential distributions in b → cτ ν decays see Refs. [25,125,[132][133][134][135][136][137][138][139][140][141][142][143]. Future measurements of b → cτ ν transitions at the Belle-II experiment will be crucial to disentangle possible new physics contributions in these decays [144].

Lepton universality tests in R M
Confirming the violation of lepton flavour universality in other b → s observables would be definite evidence in favour of new physics at work. Examples of such additional observables are R M , with M = K * , φ [145,146], defined analogously to R K , with q = d, s for M = K * , φ. 7 The expected values for R K , R K * and R φ within each bin are strongly correlated, except for the fact that hadronic uncertainties are mostly independent (but small). From the results of the fit, we find the following expected ranges for the different ratios: where it is understood that a strong (positive) correlation exists among all the predictions, lower values of one observable corresponding to lower values of another and viceversa.

Direct searches for new states at the LHC
In this model we expect a plethora of new states lying at the TeV scale: scalar bosons (in the CP-conserving limit we would have two CP-even Higgs bosons, one CP-odd Higgs and one charged scalar, cf. Section 2), heavy fermions and the massive vector bosons W , Z .
The heavy vector-like leptons will be pair-produced at the LHC via Drell-Yan processes due to their coupling to the massive electroweak gauge bosons. These will decay into gauge bosons and charged leptons or neutrinos. Though no dedicated searches for vector-like leptons have been performed at the LHC, one can obtain limits on their mass and production cross-section by recasting existing multilepton searches [159]. It was found that current limits for a heavy lepton doublet decaying to = e, µ flavours are around 450 GeV while for decays into τ -leptons the limits are around 270 GeV [159]. Searches for pair production of heavy vector-like quarks at the LHC focus primarily into final states with a third generation fermion and bosonic states, setting upper limits on the vector-like quark masses ranging from ∼ 700 GeV up to ∼ 1 TeV [160][161][162][163][164][165].
The massive vector bosons W , Z couple predominantly to the third fermion generation. The LHC phenomenology of this type of states has been discussed in Ref. [81]. The Z coupling to muons is found to be at most ∼ 12% of its coupling to τ -leptons. In the quark sector, the Z coupling to the second quark generation is found to be at most ∼ 36% of the coupling to third generation quarks. The Z boson would be produced at the LHC via Drell-Yan processes due to its coupling to b-quarks and s/c-quarks.
The total Z width normalized by the Z mass (Γ Z /M Z ) is found to grow with M Z , sinceĝ and M Z are positively correlated. Assuming that the Z can only decay into the SM fermions we have where we have neglected fermion mass effects. We obtain that Γ Z /M Z is between 2% and 31%, with Γ Z /M Z 10% for M Z 1 TeV. If kinematically open, additional decay channels of the Z boson would reduce the branching fractions to SM particles by enhancing the total Z width, making the Z resonance broader. The latter scenario will generically be the case provided the vector-like fermions are light enough, opening decay channels of the Z boson into a heavy vector-like fermion and a SM-like fermion or into a vector-like fermion pair. The decay rate for these processes is given by: Here λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + yz + xz), N C = 3(1) for (un)coloured fermions and x i = m 2 i /M 2 Z . We have denoted by F i a generic heavy fermion and by f j one of the SM-like fermions. The matrices Σ and Ω Q,L have been defined in Eqs. (52) and (53). The Z decays into a heavy fermion and a SM-like fermion are accidentally suppressed due to the small entries of the Σ matrix within the parameter region of interest. These decays therefore give small contributions to the total width in general. The decays into a pair of heavy fermions, on the other hand, can give a significant contribution to the total Z width when kinematically allowed. For instance, if the masses of the heavy leptons lie around 450 GeV we obtain a contribution to Γ Z /M Z from the decays Z → E iĒi , N iNi (i = 1, 2) of about 20% for M Z ∼ 1.2 TeV, making the Z boson a very wide resonance in this case: The ATLAS and CMS collaborations have searched for a resonance in the τ + τ − channel at √ s = 8 TeV [166][167][168][169]. Among these, the strongest limits are those coming from ATLAS and they place important bounds on the model. We have evaluated the Z production crosssection at the LHC using MadGraph (MG5 aMC 2.4.2) [170]. We find that it is possible to exclude the low-mass region where the Z resonance remains reasonably narrow and there is not much room for additional decay channels giving large contributions to the total width. The latter would require having very light exotic fermions, entering in conflict with direct searches for these states at colliders. In the heavy Z mass region ( 1 TeV) the Z resonance becomes wide (Γ Z /M Z 10%) and the interpretation of the current experimental results based on the search of a relatively narrow resonance is not valid anymore. Dedicated searches at the LHC for a broad resonance in the τ + τ − channel within the mass range ∼ 1 − 1.7 TeV would then be needed in order to test this scenario. 8 The proposed scenario also gives some predictions in the scalar sector relevant for collider searches. Neglecting mixing between the scalar bidoublet Φ and the Higgs doublets φ ( ) , the scalar spectrum will contain a heavy CP-even neutral scalar transforming as an SU(2) L singlet originating from Φ. We will denote this state by h 2 . The mass of this scalar is expected to be around the symmetry breaking scale u ∼ TeV. The dominant interactions of h 2 are with the heavy fermions and the heavy gauge vector bosons, these are described by The production of h 2 at the LHC is dominated by gluon fusion mediated by the heavy quarks and is determined by the same parameters entering in the low-energy global fit. At the centre-of mass energy √ s the production cross-section reads Here c gg represents a dimensionless partonic integral which we estimate using the set of parton distribution functions MSTW2008NLO [171] evaluated at the scale µ = M h 2 . In writing the decay rate for h 2 → gg we have taken the local approximation for the fermionic loops. For M h 2 ∼ 1 TeV, and restricting the rest of the parameters to the region described in Eq. (82), we obtain σ(pp → h 2 ) 110 − 290 fb at √ s = 13 TeV centre-of-mass energy. For M Z ∼ 1.7 TeV (and M h 2 ∼ 1 TeV) the production cross-section converges towards ∼ 110 fb. The interactions of h 2 in Eq. (88) will induce loop-mediated decays into gluons (which will hadronize into jets) and electroweak gauge bosons W + W − , ZZ, γγ, Zγ. Assuming negligible tree-level decays, the h 2 boson will manifest in this case as a very narrow resonance decaying mainly into a pair of jets. The current experimental sensitivity for dijet-resonances at the LHC around this mass range (M h 2 ∼ 1 TeV) is at the level of 10 3 fb [172,173]. The decays into electroweak gauge bosons are found to be subdominant and for M Z ∈ [1, 1.7] TeV we have: Note however that in the case where some of the heavy fermions are below the threshold M h 2 /2, tree-level decay of h 2 into these fermions becomes kinematically open and will generically dominate over the loop-induced decays commented above.

Conclusions
We have performed a phenomenological analysis of a renormalizable and perturbative gauge extension of the Standard Model. We took into account flavour observables sensitive to treelevel new physics contributions as well as bounds from electroweak precision measurements at the Z and W pole. More specifically, we have analysed the model in light of the current hints of new physics in b → c ν and b → s + − semileptonic decays, finding that the flavour anomalies can be accommodated within the allowed regions of the parameter space.
As derived from the phenomenological analysis, strong hierarchies in the flavour structure of the Yukawa couplings are required in order to accommodate both b → s + − and b → c ν anomalies. We have taken a phenomenologically oriented approach in this work, not invoking any flavour symmetry behind such structure. One interesting question would be the exploration of possible flavour symmetries accommodating the observed flavour structure. We confirm the conclusions of Ref. [4] regarding the importance of suppressing gauge bosons mixing. This translates in a tuning of tan β. Such accidental tuning would be more satisfactory if there was a dynamical or symmetry-based explanation behind. These last points also bring us to the question of the validity of our analysis, based on tree-level new physics effects, once quantum corrections are considered. These corrections might alter the flavour structure of the theory, remove accidental tunings which hold at the classical level as well as introduce new constraints from loop-induced processes such as b → sγ. Though such analysis lies beyond the scope of our work, it would be relevant in order to establish the viability of the proposed framework if the present deviations in b → s + − and b → c ν are confirmed in the future.
From the model building point of view, there are many open questions which we have not addressed in this work and would deserve further investigation, one of them being the implementation of a mechanism for the generation of the observed neutrino masses and lepton mixing angles. Our model also lacks a dark matter candidate, motivating the extension of our framework. It would be interesting to pursue the investigation of possible embeddings of the model within a larger gauge group, where the mass of the heavy fermions arise from spontaneous symmetry breaking.

A.1 Tadpole equations
The vev configuration introduced in Section 2 leads to three minimization conditions or tadpole equations. In the following we will consider all the parameters in the scalar potential to be real. Defining these are These three conditions can be solved for the mass squared parameters m 2 φ , m 2 φ and m 2 Φ .

A.2 Scalar mass matrices
The neutral scalar fields can be decomposed as Since we assume that CP is conserved in the scalar sector, the CP-even and CP-odd states do not mix. In this case, one can define the bases which allow us to obtain the scalar mass Lagrangian The mass matrix for the CP-even scalars is given by with Again, one can find two vanishing eigenvalues in M 2 H ± after applying the tadpole equations in Eqs. (92). These correspond to the Goldstone bosons eaten-up by the W and W gauge bosons.

B Pseudo-observables for Zand W -pole observables
In our model, the pseudo-observables considered in Ref. [84] are given by: where δv = −ζ 2 1 2 The family index i for these shifts covers the three fermion generations except for δg Zu i R , for which i = 1, 2. We neglect corrections to the right-handed Z and W couplings that are suppressed by the fermion masses, see Section 3. We also neglect loop contributions, which we estimate to be comparable to the tree-level contributions for ζ 0.02. However, the resulting δg's in that case would be below the limits quoted in [84].