Non-minimal $331$ model for Lepton Flavour Universality Violation in $b\to s\ell\ell$ decays

331 models constitute an extension of the Standard Model (SM) obtained by enlarging the SM gauge group $SU(3)_\text{C}\times SU(2)_\text{L}\times U(1)_Y$ to the group $SU(3)_\text{C}\times SU(3)_\text{L}\times U(1)_X$. We investigate how a non-minimal 331 model may embed lepton flavour universality violating contributions to $b\to s\ell\ell$ processes without introducing lepton flavour violation, as suggested by the recent LHCb measurements of the ratios $R_K$ and $R_{K^*}$. We discuss the model-independent scenarios of New Physics in $b\to s\ell\ell$ currently favoured by the data that could be accommodated by this model and consider a few phenomenological constraints on this model.

(1.1) where R K ( * ) are measured over specific ranges for the squared di-lepton invariant mass q 2 (in GeV 2 ), whereas R D ( * ) deals with the total branching ratios. It is interesting to make a comparison between the experimental and theoretical values for these quantities: In the experimental data the first errors are statistical and the second ones systematic. Prominent contributions to these ratio determinations have been given by Babar, Belle and LHCb [5,6,[11][12][13][14][15]. Although it is still not excluded that the previous disagreements might be accounted to statistical fluctuations of the data, or to a possible underestimate of the theoretical errors, an interesting aspect of these anomalies lies in the fact that they all seem to point in the direction of a possible Lepton Flavour Universality Violation (LFUV) in the interactions mediating the processes. Moreover, another LFU ratio has been measured recently, corresponding again to the quark decay b → c ν [16]: around 2σ above the SM predictions.
For what concerns the R K and R K * anomaly, the situation becomes even more intriguing for three reasons. First of all, the process is mediated by a Flavour Changing Neutral Current (FCNC). Since such a current cannot arise at tree level in the SM, the suppression due to the loop structure implies that the possible contribution of NP effects might arise in a significant way in this process. Furthermore, it can be shown that in the ratios R K ( * ) the hadronic uncertainties cancel to a very large extent [8,[17][18][19][20][21][22][23] 1 , thus reducing substantially the uncertainty on the theoretical expectations. Finally, these deviations concerning the branching ratios are only a part of the anomalies observed in b → sµµ decays. Contrary to b → c ν transitions, there are many other observables that have been measured, especially concerning the angular distribution of the decay products in the decays B → K * (→ Kπ)µµ and B s → φ(→ KK)µµ, and some observables (the so-called P 2 and P 5 [24][25][26]) have featured deviations from SM expectations in addition to the LFUV ratios quoted above [27][28][29][30][31]. Many model-independent analyses of these anomalies in b → s have already been performed in terms of effective field theories corresponding to the SM at the b-quark mass scale, supplemented with the additional lowest dimensional non-SM operators [18][19][20][21][22][23][32][33][34][35][36]. They are able to accommodate all the deviations observed in b → s in terms of a significant shift of the short-distance Wilson coefficient C µ 9 , possibly together with shifts in other Wilson coefficients such as C µ 9 or C µ 10 . Remarkably, the same shift is needed to explain the anomalies in the angular observables in B → K * µµ and the LFUV ratios of branching ratios R K ( * ) .
While model-independent analyses are powerful tools to understand the pattern of the anomalies in terms of NP contributions already felt at low energies, they are not able to provide a dynamical explanation for these deviations. This requires us to choose specific scenarios of physics Beyond the Standard Model (BSM) and try to see if they allow for such anomalies. Several models have been proposed to account for R K ( * ) and R D ( * ) simultaneously. Most of the successful candidates can be cast in two sets. One set includes models that try to reproduce the presence of LFUV by assuming that the relevant processes are mediated by leptoquark particles [37][38][39][40]. In the other set the process is mediated by heavy exotic gauge bosons, whose couplings depend on the generation. In this article, we analyse a model falling in the latter category, and corresponding to a specific version of the so-called 331 models [41]. 331 models constitute one of the simplest extensions of the SM. The gauge group is extended from the SM gauge group SU (3) C × SU (2) L × U (1) Y to the group SU (3) C × SU (3) L × U (1) X . These models experience thus two stages of breaking: at a heavier scale Λ N P , the extended group is broken down to the SM gauge group, for which electroweak symmetry breaking occurs at the lower scale Λ EW . Phenomenologically, these models feature heavy gauge bosons (W , Z ) as well as an extended Higgs sector triggering the two spontaneous breakdowns, leading to heavy scalar/pseudoscalar bosons (H, A), with electric charges depending on the implementation of the model.
In the most studied version [42][43][44][45][46][47][48], one simply extends each SU (2) L doublet to one of the two fundamental representations of SU (3) L , namely either 3 or3, without introducing any additional family. Furthermore, this assumption is taken together with the requirement of cancellation of chiral anomalies, that prescribes that the number of triplets is equal to the number of antitriplets. The three lepton families are then forced to belong to the same fundamental representation of the group, hence implying the family-independence of the couplings with gauge bosons. This in turn prevents any LFUV at the level of the gauge couplings to the leptons.
Another version of the 331 model, partially analysed in Refs. [49,50], extends the lepton sector by introducing two additional generations. With this assumption, one ends up with a lepton generation that transforms differently compared to the others, and hence presents different couplings with the gauge bosons; this situation suffices to guarantee the presence of LFUV. Two, rather than one, additional lepton generations, are required to preserve anomaly cancellation. We will focus on this version of the 331 model, and we will study if it can reproduce the anomalies observed in b → s processes under simple assumptions: LFUV is present and dominated by neutral gauge boson contributions, there is no significant Lepton Flavour Violation (LFV) of the form b → s 1 2 , the model should not yield too large contributions to B sBs mixing. It turns out that the model is then able to reproduce scenarios with large contributions to (C µ 9 , C µ 10 ) in good agreement with global fit analyses of b → s .
The paper is organised as follows: in section 2 we review the main features of our model, and justify our choices compared to the minimal 331 models more often studied in the literature. In section 3 we analyse the gauge boson-mediated contributions arising for the process b → s , pointing out the arising of LFUV in the couplings. In section 4 we compare these contributions with the global analyses performed in Refs. [18,19]. In section 5, we examine other simple phenomenological constraints on the model for the gauge boson contributions considered here, in particular B sBs mixing. In section 6 we conclude and discuss further extensions of the model, for instance concerning LFUV in R D ( * ) . Finally, the appendices are devoted to various computations concerning the spectrum and couplings of our model.

Features of the 331 model
Starting from the gauge group SU (3) C × SU (3) L × U (1) X (with gauge couplings g S , g, g X ), the model will undergo two Spontaneous Symmetry Breakings (SSB). The first one occurs at an energy scale Λ NP and allows to recover the SM gauge group. The subsequent one, at energy scale Λ EW , reproduces the Electroweak Symmetry Breaking (EWSB) of the SM. We assume that Λ NP Λ EW , and introduce a parameter = Λ EW /Λ NP keeping track of the order of magnitude of the NP contributions of the model.
When enlarging the SM gauge group, embedding it into the broader SU (3) C ×SU (3) L × U (1) X group, there are a few general requirements to be obeyed: • the model should contain representations consistent with the SM quantum numbers and should have no anomalies, which sets powerful constraints on the choice of representations for the fermions [45], • it should exhibit a Higgs sector able to trigger the two stages of spontaneous symmetry breaking (breaking down to the SM group and electroweak symmetry breaking) and to generate masses with a hierarchy in agreement with the observations (no light particles apart from the SM ones) [46].
For our particular purposes, we will also require that the lepton generations are not embedded equally into SU (3) L representation, in order to be able to generate LFUV at the level of the interactions.

Choice of β
We start by discussing the generators of the SU (3) L group and its connection with the SM gauge group. Leaving aside the case of SU (3) C , that presents no differences with respect to the SM, the generators of the SU (3) L gauge group are indicated withT 1 · · ·T 8 . Since the generator of the U (1) X group must commute with the generators of SU (3) L , it has to be be proportional to the identity in the space referred to the representation of SU (3) L . The normalisation of the generators is Tr[T iT j ] = δ ij /2, and 1 = diag(1, 1, 1) is the identity matrix. We define the U (1) X generator asT 9 = 1/ √ 6, since this definition implies the same normalisation relation as the other eight generators.
We can then identify the hypercharge operatorŶ in terms of the generators of the new gauge group, by requiring thatŶ commutes with all the generators of SU (2) L , which forces it to have only terms proportional toT 8 and to the U (1) X generator. Naming X the quantum number associated with U (1) X , we definê . Withλ i we indicate the Gell-Mann matrices. With this definition of the hypercharge, the electric charge operator readŝ whereT 3 = 1/2λ 3 = 1/2 diag(1, −1, 0). The electric charge is defined in general as a linear combination of the diagonal generators of the group, where the value of the proportionality constant a and β distinguishes different 331 models. In order to obtain isospin doublets which embed SU (2) L ×U (1) Y into SU (3) L ×U (1) X , we set a = 1. The way in which the SM electroweak gauge group is embedded in SU (3) L × U (1) X is encoded in the parameter β, which controls the relation between the hypercharge and theT 8 generator of SU (3) L . In order to restrict β we could demand that no new particle introduced in the model has exotic charges (i.e. different from the SM ones). Let us see how this condition operates when fermions belong to a triplet or an anti-triplet of SU (3) L . After the first stage of symmetry breaking at the scale Λ NP , the SU (3) L × U (1) X representations of the fermions are broken down to SU (2) L × U (1) Y representations as follows As just shown in Eq. (2.3), both the triplet and the anti-triplet representation of SU (3) L are broken down to a doublet plus a singlet of SU (2) L . Let us consider the case of the quarks. We will choose to identify the first two components of the triplet (or anti-triplet) with the SM doublet: their charges acquire the SM values only by setting the U (1) Y hypercharges to the SM values, that is ±β/ √ 3 + 2x = 1/3. The last entry of the triplet (or antitriplet) will be an additional, massive, fermion (called "exotic" in the following), with an electric charge ∓ √ 3/2 β + 1/6, that becomes either 2/3 or -1/3 only if we choose β = ∓1/ √ 3 2 . One can easily check that the same discussion also holds in the case of the leptons, with a similar outcome [42].
In this work, we will pick the particular value It can be related to the choice β = 1/ √ 3 by changing all the representations for their conjugates, and taking the opposite sign for the U X (1) charges. We will thus have the following definition of the electric charge operator

Fields and representations
In the following, we label the SM fermions with lower cases and the exotic ones with upper cases, choosing letters recalling their electric charge assignments. Using the notation (SU (3) C , SU (3) L , U X (1)) while referring to the representations of the particles, we introduce the following fermionic content, which ensures the cancellation of the anomalies but allows for different representations for the three lepton generations, and thus potential LFUV effects (see also App. A for a summary of the representations chosen).
For the left-handed components, we introduce [47][48][49][50] • three generations of quarks ; (2.6) • five generations of leptons . The superscripts refer to the charge and the chirality of the fields. No positively charged leptons have been introduced in the triplets. Indeed, they would only appear in L 5 , but we identify them with the charge conjugate of the right-handed component of E − 4 and e − 3 . This identification avoids the presence of charged exotic particles with masses of the order of the electroweak scale, which have not been observed 3 .
For the right-handed components, we do not consider right handed partners for neutral particles, since they would be pure singlets with respect to the whole gauge group and of no relevance in our analysis (they should be added to discuss the neutrino mass matrix, which is beyond the scope of this article). We define • the quark fields • the charged lepton fields As already indicated, the right-handed parts of e − 3 and E − 4 are not singlets, but belong to the lepton triplet L 5 . This particle content enables the cancellation of chiral anomalies. For instance, as discussed in Sec. 1, it is easy to see that the number of left-handed fermion triplets is equal to the number of left-handed fermion anti-triplets (taking into account that the quark fields are counted three times more than the lepton ones due to colour). Minimal 331 models also exhibit the anomaly cancellation by having different SU (3) L representations for the three quark generations, but having the same representation for the three lepton generations prevents these minimal models from exhibiting LFUV. More details on the requirements imposed by the cancellation of anomalies can be found in App. C.
It proves easier to discuss the spectrum of the theory after introducing the flavour vectors gathering fields with the same electric charge (for simplicity, we leave out the neutrino fields) (2.10) We also group the SU (3) L gauge bosons as 3 We discuss the structure of the fermion masses derived from the Yukawa interactions between scalar and fermions in App. B, and in particular the masses of the charged leptons in App. B.4.
and introduce The values of the charges of the V µ and Y µ bosons depend on the value of β (indeed, in the case β = 1/ √ 3, we would have V 0(0 ) µ and Y ± µ ). Let us observe that for β = 1/ √ 3, W 4,5 are both eigenstates of the charge operator with 0 eigenvalue, which allows the choice to use them, rather than Y 0(0 ) as independent degrees of freedom. We gather the interactions between the gauge bosons and the charged fermions in App. D.
Summarising, we have chosen the particle content of the model in a way that allows LFUV, but otherwise departs from the SM as little as possible. Fixing β = −1/ √ 3 ensures non-exotic charges for both SM and new fields in the spectra. Accommodating left-handed quarks and left-handed leptons in triplets or anti-triplets of SU (3) L representations, while assuming anomaly cancellation and LFUV simultaneously, forces an unequal number of quark families and lepton families. We have allowed the new degrees of freedom to be completely general, exception done for an identification in the fifth lepton generation and the exclusion of right-handed partners for neutral particles, as justified above. This last assumption implies that no Dirac mass terms can be built for neutral particles (i.e., neutrinos).

Symmetry breakings and spectrum
We are now in a position to discuss the two stages of symmetry breaking which will be assumed to be triggered by (SU (3) C singlet) scalar fields acquiring non-vanishing vacuum expectation values, in a way analogous to the SM. On the other hand, we remain as general as possible for the representations under SU (3) L , thus allowing for several scalar fields with different representations. The overall pattern of SSB is the following The SU (3) L symmetry breaking is accomplished through a triplet χ and a sextet S 1 . The EWSB is accomplished by means of two triplets η, ρ and two sextets S b,c . Details on the structure of the vacuum expectation values of these fields and on their quantum numbers can be found in App. B. There are five gauge fields that acquire a mass of the order of Λ NP , whereas the three remaining gauge fields will become massive at the electroweak scale. At the first SSB, the neutral and charged gauge bosons, W 4,5 and V ± , acquire a mass, whereas the two neutral gauge bosons X, W 8 yield a massive neutral gauge boson Z and a massless one B, with a mixing angle θ 331 : The angle θ 331 can be found by singling out the Z field in the sector of the Lagrangian including the masses of the gauge bosons, which stems from the covariant derivative in the Higgs Lagrangian. It yields sin θ 331 = g

18
, . (2.14) At the first stage of SSB, the mixing among neutral gauge bosons only involves X and W 8 , but not W 4,5 since these two classes of fields do not show the same EW quantum numbers, which correspond then to the unbroken part of the group. This can be seen for instance acting on them with the generator T 3 . After the EWSB, only the neutral gauge boson identified with the photon remains massless, consisting of an admixture of B and W 3 described by the weak angle θ W . The two mixing angles obey the relation [42] tan This is actually a very general feature of the 331 model, which can be written as cos θ 331 = β tan θ W , with a deep relation with the pattern of EWSB (see for instance Eq. (2.28) in Ref. [51] where the mixing angle is shifted by 90 • with respect to our notation). In particular, it is possible to write [42] g 2 As sin 2 θ W is close to 0.25, the perturbativity condition imposes significant constraints on the range of validity of the 331 models in the case of β = ± √ 3: the SU (3) L symmetry breaking must occur at most at a few TeV [52]. This problem of perturbativity does not affect our case β = −1/ √ 3, allowing our model to have room for a higher scale of SU (3) L symmetry breaking and significantly heavier gauge bosons, and providing a good justification to expansions in = Λ EW /Λ NP .
While the photon consists of an admixture of the W 3 and B fields only, the neutral gauge boson Z that acquires mass from EWSB includes additional components from the Z and W 4 fields. Nevertheless, the diagonalisation of the neutral gauge boson mass matrix after both stages of symmetry breaking shows that the components along the exotic fields are suppressed by 2 or higher. We will see in the following that the Z contribution to b → s involves a b → s transition already suppressed by 2 , and we will neglect the additionally 2 -suppressed contributions to the transition coming from the Z and W 4 components of the Z mass eigenstate (which we will treat as consisting only of W 3 and B at this order). The most general Yukawa Lagrangian that can be built with the scalar fields provides a (heavy) mass to all the exotic particles after the SU (3) L SSB, in agreement with phenomenological expectations. The mass matrices arising for the charged fermions after the two SSBs are discussed in Apps. B.3 and B.4. Performing a singular value decomposition of the up-type and down-type mass matrices yields the definition of the unitary rotation matrices relating (unprimed) interaction eigenstates and (primed) mass eigenstates Due to the presence of the exotic fermions, these flavour matrices are 4 × 4 (for up-type quarks) or 5×5 (for down-type quarks) unitary matrices. If we perform this diagonalisation order by order in , we observe the following pattern for the mixing matrices V (u,d) and W (u,d) • at order 0 , the SM fields are massless and they only mix among themselves; the massive exotic particles mix also only among themselves; • at order 1 , there is only mixing between SM and exotic particles; • the 2 correction yields a mixing among all the particles of the same flavour vector.
This particular structure can be understood by diagonalising the mass matrix using perturbation theory in powers of . Since the mass matrix for the SM particles is zero at O( 0 ), all SM particles are massless and degenerate at this order and they mix among themselves, whereas (heavy) exotic particles also mix among themselves. The normalisation of the eigenvectors require on the other hand that the O( 1 ) correction to an eigenvector is orthogonal to its O( 0 ) expression, leading to a O( 1 ) correction to the rotation matrix that mixes SM and exotic fields (but not SM fields alone, or exotic fields alone). A remark is in order regarding the structure of the CKM matrix. This is given by the W + coupling with quarks, which can be written as (see Eq.
with the 4 × 5 equivalent of the CKM matrix Despite V (u,d) being unitary, the presence of V yields a non-unitary V CKM in the 331 model. If we want to adequately reproduce the SM, we should however recover a unitary CKM matrix if we remain at low energies (i.e. leading order in ) and consider only the flavour subspace of the SM particles. As indicated above, at this order, the diagonalisation of the fermion mass terms occurs in a block-diagonal way: the mixing matrices V (u) and V (d) consist in two unitary blocks, one mixing the SM particles among themselves, and the other one mixing the exotic ones among themselves. Furthermore, V reduces to 1 3×3 in the SM flavour subspace. Therefore, at leading order in , the 3 × 3 SM block of V CKM will stem from the product of the two unitary 3 × 3 SM subspaces of V (u) and V (d) , ensuring that it is unitary at this order (this obviously does not mean that V CKM remains unitary at all orders in , and this 331 model does indeed generate small deviations of unitarity for V CKM ). A similar discussion could be held in the lepton sector, with the singular value decomposition of the charged lepton mass matrix leading to the definition of 5×5 unitary rotation matrices between interaction and mass eigenstates The PMNS matrix can be built by combining unitary rotation matrices V (e) and V (ν) . A discussion of the PMNS matrix would require a discussion of the neutrino spectrum, which is outside the scope of the present article.
3 NP contribution to b → s

Setting the problem
Having introduced a non-minimal 331 model with a SSB pattern leading to a phenomenologically viable spectrum, we will now investigate the consequences of the different representations for the lepton fields for LFUV in b → s . We want to determine if this model is able to reproduce the pattern of deviations indicated in the current global analyses of this rare decay [32][33][34][35][36]. These analyses are performed in the framework of the effective Hamiltonian at the b-mass scale, separating short-and long-distance physics between Wilson coefficients and local operators [53,54]: The main operators of interest for this discussion are the following: The analyses of the b → s observables (both LFUV observables and angular observables for b → sµµ and b → s ) point towards the fact that the pattern of deviations observed is consistent with a large NP short-distance contribution to C µ 9 (around 1/4 of the SM contribution). More generally, scenarios with NP contributions in C µ 9 only, in (C µ 9 , C µ 10 ) or in (C µ 9 , C µ 9 ) are particularly favoured. On the other hand, the LFUV observables agree well with the absence of significant NP contributions to any electronic Wilson coefficients C e i . Other operators suppressed in the SM, but allowed in NP models, could also be considered, in particular scalar and pseudoscalar operators. However, it turns out that these additional operators are not needed to improve the accuracy of the fit to the data (in other words, their fitted values are constrained to remain small) [18,19,32].

Gauge boson contributions
In view of these elements, we will assume that the complex pattern of EWSB of our 331 model in the scalar potential ensures that the scalar/pseudoscalar contributions to b → s are small and we will focus on the vector/axial contributions which will be assumed to be the larger ones. These contributions can only come from the neutral gauge bosons Z , Z, A, W 4,5 . We will consider contributions at the lowest order in only, and we will focus only on the non SM contribution to the Wilson coefficients (in other words, from now on C i = C N P i ). Let us start with the interaction of Z and Z with the right-handed quarks. These interactions are proportional to the identity in flavour space (see Eqs. (D.5) and (D.6)), so no flavour change can arise, at any order in . We conclude that Z and Z, do not contribute to C 9,10 in the process b → s + − . Only contributions to C 9,10 are possible.
In the case of the heavy gauge boson Z , a O( 2 ) suppression compared to the SM contribution comes from the heavy mass in the propagator of the gauge boson. The restriction of the interaction matrix to the SM particles is not proportional to the identity matrix in the interaction eigenbasis, as it can be seen in Eq. (D.8). Therefore, the flavour-changing transition b → s mediated by Z arises already after reexpressing the interaction in the mass eigenbasis using the leading order 0 rotation matrix. The overall suppression of the Z contribution is thus O( 2 ). Starting from Eq. (D.8), reexpressing the flavour eigenstates in the multiplets Eqs. (2.10) in terms of mass eigenstates, and eliminating the coupling g by means of Eq. (2.15), we can rewrite the leading-order Z contribution in terms of effective operators as where the indices k, l refer to the SM generations of the quark mass eigenstates (assuming k = l), while i, j refer to the SM lepton mass eigenstates (either from the same or different generations). The effective operators O klij 9,10 are defined exactly as in Eq. (3.2), corresponding to the (q k q l )(¯ i j ) flavour structure. The fine-structure constant is α = e 2 /(4π). The V and W matrices provide the mixing matrices arising from the diagonalisation of the EWSB mass terms in the subspace of left-handed and right-handed SM fields. We stress that these rotations are related but cannot be identified with the CKM or PMNS matrices and they can be considered only at order 0 for our purposes (we have exploited their unitarity at that order for the δ ij contributions). We notice that the presence of the mixing matrices yields LFUV couplings, and moreover a leptonic i = j contribution might arise, corresponding to lepton-flavour violating transitions b → s + − , with different leptons in the final state, = , which is a frequent feature of models generating LFUV couplings [55].
In the case of the SM gauge boson Z, there is no b → s transition allowed at order 0 , since the 3 × 3 unitary rotation matrices restricted to the SM subspace cancel, following the same arguments as the discussion of the unitarity of the CKM matrix at the end of Sec. 2.3. The transition does not arise at order 1 either, since there is no correction to the mixing between SM particles at this order. The mixing between SM particles, leading to potential FCNC currents, starts only at order O( 2 ). Since there is no suppression due to the mass of the intermediate gauge boson here, we conclude that the NP contribution from the SM gauge boson Z starts at O( 2 ), the same order as the Z contribution, although for different reasons. Indeed, starting from the interaction eigenbasis in Eq. (D.9) and switching to the mass eigenstates, we can express the part of interaction relevant to the process as correction to the rotation matrix V (d) between interaction and mass eigenstates for the left-handed down sector. As stated earlier,V (d) mn = 0 if m and n are both SM or both exotic, which means that the sum over λ is restricted to exotic components here (as k, l are SM components). Since the NP quark coupling to the Z gauge boson is already of order O( 2 ), we need only the O( 0 ) coupling to the charged leptons. Due to the unitary block structure of the mixing matrix at this order and the structure of the Z coupling to SM leptons (proportional to identify), we see that the rotation matrices cancel out, leading to the diagonal structure indicated in the leptonic sector of Eq. (3.4). In terms of effective operators and adopting the same notation of Eq. (3.3), Eq. (3.4) can be rewritten as We observe that the coupling is the same for all the light leptons, i.e. non-universality does not arise at order 2 in the interaction with Z. By comparing Eq. (3.3) and Eq. (3.6), we explicitly see that although the non standard coupling originated from the Z boson are suppressed of order 2 with respect to the ones of the Z boson, the contributions are the same order, due to the additional 2 suppression due to the Z propagator.
There are no further contributions to be considered from the other neutral gauge bosons. Indeed, for the photon A, we see from Eq. (D.7) that the interaction with down-type quarks is proportional to the identity matrix in flavour space, so that there are no FCNC from the photon interaction. Concerning W 4,5 , we see from Eqs. (D.3) and (D.4) that these gauge bosons always couple a SM particle with an exotic one in the interaction basis. In order to obtain a W 4,5 -mediated b → s, we need to consider the interaction with one of the exotic interaction eigenstates, which will contain a SM mass eigenstate due to the rotation matrix V (d) . As indicated earlier, this occurs only at order O( ). Furthermore, the process is mediated by a heavy gauge boson, adding a further O( 2 ) suppression. Therefore the W 4,5 contributions to the process are of order O( 3 ) and can be neglected compared to the O( 2 ) NP contributions from Z and Z gauge bosons.

Wilson coefficients and lepton-flavour violation
The joint effect of the two O( 2 ) contributions from Z and Z processes in our 331 model can be rewritten introducing the quantities where θ 331 and g X have been expressed in terms of θ W and g by using Eq. (2.15). In order to focus on b → s transitions, let us set the quark indices to k = 2 and l = 3 and rename coefficients and operators by removing the corresponding labels. We get where the operators O ij 9,10 denote operators with given lepton flavours i, j, with the same normalisation as in Eq. (3.2). We obtain the following NP contributions to the Wilson coefficients We see that LFUV contributions arise from the Z contribution, whereas the Z contribution does not depend on the lepton flavour. In addition to the violation of lepton-flavour universality, our model allows for lepton-flavour violation, such as b → s + − for = .
However, since there have been no experimental indications of such processes up to now, we will assume that these processes are suppressed, and for simplicity, we will set these coefficients to 0 when the two lepton indices are different, for any i = j. Imposing this, we get the system The trivial solution f Z = 0 has to be discarded, since it would remove the only source of LFUV, i.e. the coupling of the charged leptons to Z . The alternative solution is In other words, we require that the left-handed interaction eigenstate of the first generation and the right-handed interaction eigenstate of the third generation are also mass eigenstates. Due to the unitarity of these 5 × 5 matrices, we have then which means that they must both stay within the [0,1] range, keeping in mind that V and W entries on the right hand side of Eqs. (3.11)-(3.12) are of order . In the following, and for simplicity of notation, repeated indices (like II or ee) will be denoted with a single index (I or e). We now consider two different scenarios: • Case A: the index I for which the rotation matrix element V 1I is nonzero is the same as the index J for which the element W (e) 3J is non-vanishing; • Case B: the two indices corresponding to non-vanishing matrix elements are different.

Case A
If we denote with J the generation for which both entries for the rotation matrices are nonzero, we get We get identical Wilson coefficients for the other two generations i = J, for which the entries in the rotation matrices vanish, (3.14) Inverting these relations we get We have now to identify whether the electron corresponds to the index J or not. As discussed in Sect. 3.1, we set to zero the corresponding NP contributions to the effective Hamiltonian, C e 9,10 , on the basis of phenomenological constraints.  Figure 1: Regions allowed for the Wilson coefficient C µ 9 and C µ 10 (abscissa and ordinate, respectively) in scenarios A (left) and B (right). The thick black intervals correspond to the 1σ interval for one-dimensional scenarios from Ref. [32].
• If we identify the electron with another index i = J (identifying the electron with a generation with vanishing entries), we must have C i 9,10 = 0. From Eq. (3.15), we obtain that f Z = 0, so that no LFUV could be generated. We have thus to discard this possibility.
• If we identify the electron with the index J (identifying the electron with the generation with a non-vanishing entry), we set the corresponding NP Wilson coefficients to zero. In this case, Eq. (3.15) yields constraints on the possible values for the muon Wilson coefficients C i 9,10 = C µ 9,10 (also equal to C τ 9,10 ): ≤ 1, these expressions yield a wedge in the (C µ 9 , C µ 10 ) plane. The constraint from λ (L) is the more stringent one, imposing the ratio C µ 10 /C µ 9 to remain between -1.75 and -1 (we use sin 2 θ W 0.235), as indicated as a grey wedge on the left-hand side of Fig. 1.
In summary, in case A, we find that the electron has to be identified with the generation with a non-vanishing entry in the rotation matrices V and W . Muons and taus give the same NP contribution to the Wilson coefficients C 9 and C 10 Eqs. 3.16, imposing that |C µ 10 | ≥ |C µ 9 |.

Case B
In case B, we have two different indices I = J such that V Inverting with respect to f Z , f Z , λ Moreover, if we denote K the remaining SM generation (K = I, J) we have the following relationships We still have not identified which of the I, J, K indices refers to the electron, muon, or tau leptons: • If we identify the electron with J, we set C J 9 = C J 10 = 0 and from the first and last relations of Eq. (3.18) we get λ (R) leading to an inconsistency, since the λ must be non-negative.
• If we identify the electron with K, we set C K 9 = C K 10 = 0 and from Eq. (3.19) we get which can be used in Eq. (3.18) to show that f Z = f Z = 0, so that this solution can be discarded.
• If we identify the electron with I, we set C I 9 = C I 10 = 0, the solutions Eq. (3.18) become -17 -from which we can read the expressions for the λ: leading to the following conditions on the non-vanishing NP Wilson coefficients We see that the value found for λ lies in the allowed interval [0, 1]. Furthermore, requiring that λ (R) J also remains in this interval yields a constraint on the Wilson coefficients: if we identify the muon with K, we have the exact equality C µ 10 /C µ 9 = −1, and if we identify the muon with J, the slope C µ 10 /C µ 9 is constrained between -1 and -0.28 (using sin 2 θ W 0.235). These constraints are indicated in grey on the right-hand side of Fig. 1.
In summary, in case B, we find that the electron generation must be identified with the non-vanishing entry I in the rotation matrices V . Two possibilities can be considered concerning the non-vanishing entry J in the rotation matrices W . If we identify J with the muon generation, muons and taus have different NP contribution for the corresponding Wilson coefficients C 9 and C 10 , imposing that |C µ 10 | ≤ |C µ 9 |, the NP contribution to C µ 10 is different from zero, and C τ 9 = −C τ 10 . If we identify J with the tau generation, one gets again different NP contributions for the Wilson coefficients C 9 and C 10 for muons and taus, the roles played by muons and taus are reversed, and thus one gets C µ 9 = −C µ 10 . Both cases yield thus NP contributions given by Eq. (3.24).

Comparison with global analyses
We perform a comparison between the 331 model contributions to the process b → s + − and the global analysis of b → s anomalies performed in Refs. [18,19,32] (similar results were obtained in recent works from other groups, see Refs. [32][33][34][35][36]). In these works, the authors pointed out scenarios in which NP contributions to the Wilson coefficients C µ 9( ),10 are favoured whereas no NP contributions occur for other Wilson coefficients (including all the electronic ones). In particular they identified three specific one-dimensional scenarios as particularly favoured: • NP in C µ 9 = −C µ 9 , with the 1σ interval [-1.18, -0.84]: this scenario cannot be described in the framework of our non-minimal 331 model, where no FCNC arise for righthanded quarks, meaning that C µ 9 = 0 (see Sec. 3.2); • NP in C µ • NP in C µ 9 = −C µ 10 , within the 1σ interval [-0.73, -0.48]. From the discussion of the previous section and Fig. 1, we see that this scenario is allowed in both scenarios A and B.
Our non-minimal 331 model appears to be able to account for the b → s anomalies observed as far as we consider the C µ 9 = −C µ 10 case. More generally, it would be able to reproduce other favoured values for the two-dimensional scenario (C µ 9 , C µ 10 ) with negative NP contributions to C µ 9 and positive to C µ 10 (see top-left plot in Fig. 1 in Ref. [32]). For simplicity and illustration of the potential of our 331 model, we will focus here on the one-dimensional (1D) scenario C µ 9 = −C µ 10 considered in Refs. [18,32]. Imposing this equality, we see that in both cases A and B we have λ (L) e = 1 − tan 2 θ W 5 and [1D] (4.1) so that NP contribution to C µ 9 is given by parameters of the 331 model included in f Z , where the only unknown quantities are M Z and V * (d) 33 . These can be further constrained by other processes, and in particular B s meson mixing, as explained in the next section.

Phenomenological constraints on Z and Z couplings
We have built our 331 model in order to generate vector/axial LFUV contributions to b → s transitions. This has led us to assume that the dominant contributions for these couplings (bs and µµ) came from the gauge bosons rather than the Higgs sector, and actually that the dominant contributions came from anomalous couplings of the Z gauge boson as well as tree-level exchanges of a Z gauge boson. Even in this restricted setting, there are additional constraints to be considered on these couplings from the phenomenological point of view, as discussed in Refs. [42-44, 56, 57].
A first class of constraints for additional contributions from neutral gauge bosons comes from the violation of unitarity in the CKM matrix. One has to consider the corrections to the decay µ − → e − ν µνe (as it defines the normalisation for all decays through G F ) as well as the decays b, s, d → ue −ν e (leading to |V ub |, |V us | and |V ud | determinations assuming the SM). This corresponds to box diagrams involving both W and Z or Z bosons, as shown in graphs (a) and (b) of Fig. 2. One can expect the Z contribution to be small, as the diagrams require to have a Z coupling to the first generation, which is suppressed in our model. On the other hand, the FCNC couplings of the Z to quarks occur (in principle) between all down-type quarks, meaning that we need a detailed understanding of the O( ) mixing matrixV (d) , see Eq. (3.4), in order to compute this correction in our model. Such a detailed knowledge might be obtained by a complete analysis of all flavour constraints on our model, which is far beyond the scope of the present article.
In our model, both Z and Z gauge boson give a tree level contribution to the B s − B s mixing, as can be seen on Fig. 2. Contrary to the previous constraint, this one focuses only   Figure 2: Gauge contributions to the violation of unitarity of the CKM matrix in the first row (for matrix elements determined leptonic and semileptonic processes) and to B sBs mixing (see Refs. [56,57]). on the bs mixing, and can provide useful information in addition to the b → s decay. As before, we restrict our discussion to contributions of order O( 2 ), borrowing from the results in Sec. 3.2. At this order, Z gives no contributions to the mixing. Indeed, the bsZ vertex has a suppression of O( 2 ), due to the structure of the unitary matrices needed to obtain physical states. The contribution to B s − B s mixing will have two such vertices, and hence be suppressed by a factor O( 4 ). Concerning the Z contribution, we only need to take into account the O( 2 ) suppression coming from the heavy gauge boson propagator, since the bs vertex for this gauge boson is already mediated at O( 0 ).
As discussed in Eq. (D.8), the relevant part of the interaction for B s − B s is thus (in the interaction eigenbasis) Expressing in terms of effective operators of eigenstates and using Eq. (2.15), one obtains where we will focus as usual on the case k = 2, l = 3.
The SM contribution to the mixing reads [58] where S is the Inami-Lim function and m t is the top quark mass defined in the M S scheme. As in Ref. [58], we take S mt 2 M 2 W 2.35, for a top mass of about 165 GeV, and η B = 0.8393 ± 0.0034, which comprises QCD corrections.
Considering the modulus of the ratio of the NP contribution over the SM, one gets In  [59,60]. For those values, we evaluate the NP contribution to the Wilson coefficient C µ 9 = −C µ 10 in the one-dimensional scenario as expressed in Eq. (4.1). The allowed values found in the scan are plotted in Fig. 3.
We see that values of C µ 9 = −C µ 10 can reach -1.8, in agreement with the results of global analyses of b → s , corresponding to r Bs = 0.1, M W /M Z = 0.3 and d −0.002. The allowed region is limited by the fact that we have numerically using Refs. [2,67], which leads to the parabolic constraint r Bs = (C µ 9 ) 2 ×0.003/(M W /M Z ) 2 ≥ 0.03 × (C µ 9 ) 2 , represented in Fig. 3. As we saw in the previous sections, our 331 model can accommodate various NP contributions to (C µ 9 , C µ 10 ). In the simple one-dimensional scenario C µ 9 = −C µ 10 , we can accommodate both B sBs mixing and b → s data, with a NP scale (and in particular a Z ) around the TeV scale. Choosing different values for (C µ 9 , C µ 10 ) would extend the parameter space for NP allowed, with the possibility to use not only the value of f Z , but also f Z , to accommodate the data.

Conclusions
Among many achievements, the LHC experiments have been able to investigate many rare flavour processes, with very interesting outcomes. In particular, the LHCb experiment has identified several deviations from the Standard Model in the b → s transitions, with interesting hints from violation of the lepton flavour universality. These deviations can be elegantly explained within model-independent effective approaches, where a few Wilson coefficients receive significant NP contributions. This has triggered a lot of theoretical work to identify viable models explaining such deviations, among which Z models and leptoquark models have been often used.
In the present paper, we try to embed a Z model in a more global extension, widely used in the literature, namely the 331 models where the gauge group SU (3) C × SU (3) L × U (1) X breaks down at a high scale into the SM gauge group, before undergoing a second transition at the electroweak scale. The minimal versions of such models do not feature lepton flavour universality violation as they have to obey anomaly cancellations. We thus investigated a non-minimal 331 model with 5 lepton triplets able to include LFUV. We described the choices made to build this model in order to have all additional gauge bosons and fermions with heavy masses (of the order of the scale of SU (3) L breaking) and electric charges similar to those present in the SM. We worked out how this model could reproduce the deviations observed in b → s transitions. This requires us to assume that the deviations are dominated by neutral gauge boson contributions (anomalous bsZ coupling due to fermion mixing as well as flavour-changing neutral coupling to a heavy Z boson). The absence of significant contribution to b → see and lepton-universality violating processes allowed us to set constraints on the mixing matrices between interaction and mass fermion eigenstates.
We identified two different cases for the mixing matrices, with a rather simple outcome. Our model turns out to have no right-handed currents, but it is able to accommodate significant NP contributions to C µ 9 (negative) and C µ 10 (positive), in agreement with NP scenarios favoured by global fits. In each case, we could make predictions concerning the τ Wilson coefficients (the electron ones being assumed to receive no NP contribution). We considered additional phenomenological constraints on Z and Z couplings in order to check the viability of our model: if the unitarity of the first row of the CKM matrix is not powerful in our case due to the large number of parameters involved, B sBs mixing proves much more powerful.
Considering these results, it would be interesting to progress further in the study of this non-minimal 331 model. Since we are able to predict in each case the values of Wilson coefficients for b → sτ τ from the electronic and muonic ones, it would be interesting to predict the deviations arising to related observables from our model, whether in decays or in B sBs mixing [61][62][63][64].
Under our simplifying assumptions (no lepton-flavour violation b → s i j , no contribution to b → see, opposite contribution to C µ 9 and C µ 10 ), we saw that we are able to accommodate both b → s and B sBs mixing observables at the price of a NP scale of order 1 TeV. Considering different values of NP contributions to C µ 9 and C µ 10 might also enable to increase the scale of NP allowed. It would also be interesting to compare this constraint with direct searches for Z bosons, taking into account the pattern of couplings specific to our model.
Moreover, it would also be natural to consider the other hints of LFUV currently present in flavour physics, namely R D and R D * . Global model-independent analyses show that the LFUV deviations seen in b → c ν branching ratios can be explained by vector/axial exchanges, whereas scalar/pseudoscalar exchanges are disfavoured [65,66]. In our model, the situation is a bit different compared to b → s transitions. Indeed the heavy charged bosons have no couplings with SM fields in the interaction eigenbasis, which means that the SM quark and lepton couplings will be induced again by mixing (each counting at O( )) and further suppressed by the heavy gauge boson mass, leading to a contribution O( 4 ). The light W ± bosons have diagonal couplings in the SM subspace in the interaction eigenbasis (see Eq. (D.2)), which means that LFUV will appear only due to mixing effects in leptons.
This effect can in principle be of order O( 2 ) or lower, depending on the structure of the mixing in the neutral lepton sector. For this reason, the deviations observed in b → c transitions could also be explained in our model through gauge boson contributions only. The discussion requires an accurate analysis of the neutrino spectrum, and we leave it for future work.
The additional requirements from R D and R D * would thus allow us to further refine our non-minimal 331 model, and to determine if it constitutes a viable alternative to explain the LFUV processes currently observed in b-decays. If it passes these tests, it could provide an interesting alternative to current NP models used to explain the deviations in b-quark decays, with a potential to be tested both through deviations in flavour processes among other generations of quarks and leptons and through direct production searches at LHC. tality during his stay where part of the present work was carried out. MM is thankful to the Erasmus Traineeship Program of Università di Napoli Federico II and Université Paris Sud, and to the Laboratoire de Physique Théorique d'Orsay for hosting her internship during which part of the work was carried out. This work received financial support from the grant FPA2014-61478-EXP and from the EU Horizon 2020 program from the grants

A Fermionic content of the model
We summarise the U (1) charges of the fermionic content of our model (for the charged fermions) in Tab. 1. We recall that the lower case letters denote light fields corresponding to the SM, whereas upper case letters correspond to heavy exotic fields. As discussed in Sect. 2.1, all fields have only charges already present in the SM.

B Higgs fields and Yukawa Lagrangian
We need to build gauge invariant terms for the coupling between a Higgs field and two fermions, so that we obtain appropriate mass terms after SSB. This constrains possible representations for the scalar fields. Since the fermions transform either as a 3 or as a3 under SU (3) L , we only have a limited number possibilities [46] for a scalar field Φ, which can only be a singlet, a triplet or a sextet 6 .
In the following, we will not analyse the possibility of a singlet scalar. Electromagnetic invariance makes it a scalar under U (1) X . Thus, after the two steps of SSB, its vacuum expectation value will never give rise to a mass term for the gauge bosons or the charged fermions, and, as indicated before, neutral leptons are outside the scope of the present work.
For the first transition 331 → 321, we can have triplet or sextet scalar fields, denoted χ, χ and S 1 respectively. In order to break neither SU (2) L nor U (1) EM invariances at this stage, the following conditions for vacuum expectation values of the Higgs fields hold which sets the v.e.v.s and U (1) X charges of the scalar fields responsible for the first SSB. We have Table 1: Fermionic content of the model and associated U (1) charges.
The Yukawa terms that can be built with the sextet are then of the form where U R represents both u R i and T R 3 . The equivalent form in the lepton sector is where L −R represents any of e −R 1,2 , E −R 1 . The lepton sector also allows combination of SU (3) L triplets and antitriplets, as where the label a can assume values 2, 3, 4 and i, j, k are indices referred to SU (3) L .
The second, electroweak, transition 321 → 31 can involve two triplets η and ρ, and sextets, denoted S i . The electromagnetic gauge invariance still holds after this SSB, which yields the following constraints on the v.e.v.ŝ In order to choose the right alignment for sextet and triplets, we start from the most general ones, impose a zero charge and verify if we can build Yukawa terms involving these scalar fields and invariant under U (1) X . The v.e.v.s of the scalar fields responsible for EWSB are The U (1) X invariant terms built with sextets arē and for the triplets, we have where we have used the same notation of the previous SSB. Therefore, the Yukawa Lagrangian is where Y d,u , y d,u , j d,u represent the Yukawa couplings introduced respectively for χ, ρ and η.
• for leptons where Y, y, K, k, f, c, J, j represent the Yukawa couplings, with a, b = 2, 3, 4 and L i = e −R 1,2 , E −R 1 , and where the i, j, k indices are referred to the SU (3) space.

B.3 Quark masses
After the two SSBs, the quark mass terms arising from the Yukawa Lagrangian read It is possible to rewrite these mass terms in the form of a matrix product with the flavour vectors D, U , introduced in Eq. (2.10) as 19) The diagonalisation in the limit v = w 1 = w 2 = 0 (before the EWSB) shows that the number of quarks that remain massless after the SU (3) L SSB is three for up-type and three for down-type quarks (for a given colour). This is exactly equal to the number of SM particles, meaning that all the new exotic particles acquire a mass of the scale Λ N P of the SU (3) L SSB. This feature of the model is required if we want to justify why such particles have not yet been observed at the electroweak scale.

B.4 Charged lepton masses
In our model, we have identified the charged elements of 5 with the charge conjugated right handed components of particles already introduced in other generations; to be more precise, we have set (B.20) Apart from limiting the number of additional degrees of freedom, the main reason of this identification is not clear until we consider the charged exotic masses.
Without such identification, the introduction of the right-handed degrees of freedom of the charged leptons appearing in the fifth generation implies the additional Yukawa term where P +R represents the right handed components of the positively charged elements E + 5 , F + 5 of 5 . Furthermore, the vector L a in (B.15) stands now for L i = e −R 1,2,3 , E −R 1,4 . Introducing the flavour vector for negatively charged leptons after the first SSB we get the following mass matrix One can check that the degeneracy of the 0 eigenvalue of this matrix is greater than 3, implying that out of all the charged leptons, not just the ones to be identified with the SM ones acquire mass at the EW scale.
As indicated in Sec. 2.2, we avoid the presence of charged exotic particles with masses of the order of the EW scale, which have not been observed phenomenologically, through the identification of the charged elements of 5 with the charge conjugates of the right-handed components of particles already introduced for other generations. With this assumption, the mass matrix of charged leptons originating after the two stages of SSB becomes [48] The diagonalisation in the limit v = w 1 = w 2 = 0 (before the EWSB) shows that the number of leptons that remain massless after the SU (3) L SSB is three. This is exactly equal to the number of SM particles, meaning that all the new exotic particles acquire a mass of the scale Λ N P of the SU (3) L SSB. This feature of the model is required if we want to justify why such particles have not yet been observed at the electroweak scale.

C Anomaly cancellation
Particularly stringent constraints for 331 model building arise from requiring that the theory is free from quantum anomalies. We list here the relations among the fermion charges that need to be satisfied. We denote with Q the quark left-handed generations, q the corresponding singlets, the leptonic multiplets and s the corresponding singlets. Imposing the vanishing of the triangular anomaly coupling to the different gauge bosons of the theory leads to [45] It is clear from Eq. (C.2) that we cannot generate LFUV couplings for the gauge bosons unless we introduce additional lepton families. Indeed, if we call N Q (NQ) the number of quark generations transforming as a 3 (3), with similar notation for the leptons , the anomaly cancellation in Eq. (C.2) yields Restricting to just three generations of quarks N Q + NQ = 3, we see that one has several possibilities. If we assume that all three quark families transform in the same way, one needs at least 9 lepton generations (3 SM leptonics and 6 exotic ones), which would then transform all in the same opposite way to get the appropriate anomaly cancellation. Since all leptons transform in the same way, there is no possibility to generate different couplings between the leptons and the gauge bosons, and thus no LFUV can arise from these couplings.
The situation changes if one of the quark families transforms differently compared to the others. Indeed, if we assume only two quark families to transform as a 3, we obtain Assuming three lepton generations implies that N = 3, N = 0. In this minimal model, often considered in the literature, there is no possibility to generate LFUV from the identical couplings of the gauge bosons to all lepton families. We can increase the number of lepton generations. Assuming four generations, i.e., N + N = 4, yields no integer solutions for Eq. (C.7). The next possibility is N + N = 5 lepton families, so that N = 4, N = 1, which provides LFUV in the gauge couplings to leptons [45]. This is the non-minimal choice that we adopt.

D Currents
We provide the expression of the couplings of the gauge bosons with the fermions, the latter being expressed in the interaction eigenbasis.

D.1 Charged currents
For the non-SM charged gauge boson V ± we get    For the SM charged gauge bosons W ± we get

D.2 Neutral currents
First we provide the interactions with the non-SM neutral gauge bosons W 4,5 , Z   For the Z contribution we isolate the relevant part, i.e. the one involving D, f − , from Eq. (D.5) and rewrite it removing the terms that are proportional to the identity in flavour space. We also restrict to the light part of each flavour vector, i.e. the one spanned by the light eigenvectors. We get A similar analysis can be carried out for the interaction with Z. The restriction of the interaction Eq. (D.6) to the light part of the fermions involved in the process reads L Z = g cos θ W Zµ − 1 + cos 2 θ 331 2D L γ µ D L + −1 + 3 cos 2 θ 331 2f −L γ µ f −L + 3 cos 2 θ 331f −R γ µ f −R . (D.9)