Explaining $h\to\mu^\pm\tau^\mp$, $B\to K^* \mu^+\mu^-$ and $B\to K \mu^+\mu^-/B\to K e^+e^-$ in a two-Higgs-doublet model with gauged $L_\mu-L_\tau$

The LHC observed so far three deviations from the Standard Model (SM) predictions in flavour observables: LHCb reported anomalies in $B\to K^* \mu^+\mu^-$ and $R(K)=B\to K \mu^+\mu^-/B\to K e^+e^-$ while CMS found an excess in $h\to\mu\tau$. We show, for the first time, how these deviations from the SM can be explained within a single well-motivated model: a two-Higgs-doublet model with gauged $L_\mu-L_\tau$ symmetry. We find that, despite the constraints from $\tau\to\mu\mu\mu$ and $B_s$--$\overline{B}_s$ mixing, one can explain $h \to\mu\tau$, $B\to K^* \mu^+\mu^-$ and $R(K)$ simultaneously, obtaining interesting correlations among the observables.


I. INTRODUCTION
So far, the LHC completed the SM by discovering the last missing piece, the Brout-Englert-Higgs particle [1,2]. Furthermore, no significant direct evidence for physics beyond the SM has been found, i.e. no new particles were discovered. However, the LHC did observe three 'hints' for new physics (NP) in the flavor sector, which are sensitive to virtual effects of new particles and can be used as guidelines towards specific NP models: h → µτ , B → K * µ + µ − , and R(K) = B → Kµ + µ − /B → Ke + e − . It is therefore interesting to examine if a specific NP model can explain these three anomalies simultaneously, predicting correlations among them.
LHCb reported deviations from the SM predictions [3,4] (mainly in an angular observable called P 5 [5]) in B → K * µ + µ − [6] with a significance of 2-3 σ depending on the assumptions of hadronic uncertainties [7][8][9]. This discrepancy can be explained in a model independent approach by rather large contributions to the Wilson coefficient C 9 [10][11][12], i.e. an operator (sγ α P L b)(µγ α µ), which can be achieved in models with an additional heavy neutral Z gauge boson [13][14][15]. Furthermore, LHCb [16] recently found indications for the violation of lepton flavour universality in which disagrees from the theoretically rather clean SM prediction R SM K = 1.0003 ± 0.0001 [17] by 2.6 σ. A possible explanation is again a NP contributing to C µµ 9 involving muons, but not electrons [18][19][20]. Interestingly, the value for C 9 required to explain R(K) is of the same order as the one required by B → K * µ + µ − [8,21]. In Ref. [15], a model with gauged muon minus tauon number (L µ − L τ ) was proposed in order to explain the B → K * µ + µ − anomaly.
Supplementing the model of Ref. [35] with the induced Z quark couplings of Ref. [15] can resolve all three anomalies from above. Interestingly, the semileptonic B decays imply lower limit on g /M Z , which allows us to set a lower limit on τ → µµµ, depending on h → µτ .

II. THE MODEL
Our model under consideration is a 2HDM with a gauged U (1) Lµ−Lτ symmetry [35]. The L µ − L τ symmetry with the gauge coupling g is broken spontaneously by the vacuum expectation value (VEV) of a scalar Φ with Q Φ Lµ−Lτ = 1, leading to the Z mass and Majorana masses for the right-handed neutrinos 1 . Two Higgs doublets are introduced which break the electroweak symmetry: Ψ 1 with Q Ψ1 Lµ−Lτ = −2 and Ψ 2 with Q Ψ2 Lµ−Lτ = 0. Therefore, Ψ 2 gives masses to quarks and leptons while Ψ 1 couples only off-diagonally to τ µ: Here Q ( ) is the left-handed quark (lepton) doublet, u (e) is the right-handed up-quark (charged-lepton) and d the right-handed down quark while i and f label the three generations. The scalar potential is the one of a U (1)-invariant 2HDM [44] with additional couplings to the SM-singlet Φ, which most importantly generates the doublet-mixing term that induces a small vacuum expectation value for Ψ 1 [35]. We define tan β = Ψ 2 / Ψ 1 and α is the usual mixing angle between the neutral CP-even components of Ψ 1 and Ψ 2 (see for example [44]). We neglect the additional mixing of the CP-even scalars with Re[Φ]. Quarks and gauge bosons have standard type-I 2HDM couplings to the scalars. The only deviations are in the lepton sector: while the Yukawa couplings Y i δ f i of Ψ 2 are forced to be diagonal due to the L µ − L τ symmetry, ξ τ µ gives rise to an off-diagonal entry in the lepton mass matrix: It is this τ -µ entry that leads to the LFV couplings of h and Z of interest to this letter. The lepton mass basis is obtained by simple rotations of (µ R , τ R ) and (µ L , τ L ) with the angles θ R and θ L , respectively: The angle θ L is automatically small and will be neglected in the following. 2 A non-vanishing angle θ R not only gives rise to the LFV decay h → µτ due to the coupling in the Lagrangian, but also leads to off-diagonal Z couplings to right-handed leptons while the left-handed couplings are to a good approximation flavour conserving. In order to explain the observed anomalies in the B meson decays, a coupling of the Z to quarks is required as well, not inherently part of L µ − L τ 2 Choosing Q Lµ−Lτ = +2 for Ψ 2 would essentially exchange θ L ↔ θ R [35], with little impact on our study. models (aside from the kinetic Z-Z mixing, which is assumed to be small). Following Ref. [15], we introduce heavy vector-like quarks, i.e. and , coupling them to the Z boson. Yukawa-like couplings involving the heavy vector-quarks, the light chiral quarks and Φ then induce couplings of the SM quarks to the Z once Φ acquires its VEV. Thus, integrating out the heavy vectorlike quarks gives rise to effective Z d i d j couplings [45,46] of the form with hermitian matrices Γ dL ij that are related to the vector quark masses m Q,D,U and Yukawa couplings Y Q,D,U as follows [15]: which holds in the approximation |Γ

III. FLAVOUR OBSERVABLES
We will now recall the necessary formula in the region of interest (i.e. small θ R ) considering only the processes giving to most relevant bounds on our model, i.e. B s -B s mixing, neutrino trident production and τ → 3µ.
The branching ratio for h → µτ reads where Γ SM 4.1 MeV is the decay width in the SM for a 125 GeV Higgs [47] and Γ h τ µ is defined in Eq. (7). Comparing this to Eq. (2) one sees that both sin θ R = 0 and cos(α − β) = 0 are required to explain the CMS excess [35].  (10). The vertical green lines illustrate the naive LHC limit | cos(α − β)| < ∼ 0.4, horizontal lines denote the 90% C.L. limit on τ → 3µ via Z exchange. Right: Allowed regions in the Γ dL 23 -m Z /g plane from B → K * µ + µ − and R(K) (yellow) and Bs mixing (blue). For Bs mixing (light) blue corresponds to (mQ = 15m Z /g ) mQ = m Z /g . The horizontal lines denote the lower bounds on m Z /g from τ → 3µ for sin(θR) = 0.05, 0.02, 0.005. The gray region is excluded by NTP.

B. Lepton decays
While the Higgs contributions to τ → µµµ and τ → µγ turn out to be very small in most regions of parameter space [35] due to the small lepton masses involved, the Z contributions to τ → 3µ can be sizable [42] and restrict θ 2 R /v 4 Φ . The branching ratio is given by which has to be compared to the current upper limit of 2.1 × 10 −8 at 90% C.L. [48] obtained by Belle. A combination with data from BaBar [49] gives an even stronger limit of 1.2×10 −8 at 90% C.L. [50], to be used in the following. For small θ R , the branching ratio for τ → µγ is proportional to the same combination θ 2 R /v 4 Φ , but highly suppressed by 2α/π, and hence not as restrictive. . 4 While in our model the contribution to C 10 is suppressed by sin(2θ R ) (or even sin(2θ L )), the Wilson coefficients C µµ 9 and C µµ 9 with muons are generated (as well as C τ τ 9 and the θ R 4 For conventions see Refs. [8,19]. is not affected, which naturally generates violations of lepton flavour universality in B → Kµ + µ − /B → Ke + e − . We find C ( )µµ 9 where we set cos(2θ R ) = 1. As already noted in Ref. [10,51] C µµ 9 < 0 and C µµ 9 = 0 gives a good fit to data. Using the global fit of Ref. [8] we see that at ( Interestingly, the regions for C µµ 9 required by R(K) and B → K * µ + µ − lie approximately in the same region. Furthermore, a good fit to the current data does not even require C µµ 9 [8], so we neglect it in the following for simplicity. This can be achieved in the limit m D m Q , resulting in Γ dL Γ dR . We will also assume our C µµ 9 to be real for simplicity. Note that our model predicts the decay B → Kµτ (recently discussed in Refs. [52]) to be suppressed by θ 2 R compared to B → Kµµ, while B → Kµe and B → Kτ e are forbidden.

D. Bs-Bs mixing
The interactions of Z and Φ relevant for B → Kµ + µ − also contribute to B s -B s mixing [15]. For m D m Q , We require the NP contribution to be less than 15% in order to satisfy the experimental bounds [15]. Due to the dominance of the vector-quark Q we can express Γ dL 23 directly in terms of C µµ 9 from Eq. (15) and find the upper bounds m Z /g < 3.2 TeV/|C µµ 9 |, m Q < 41 TeV/|C µµ 9 | . (18) Combining Eq. (18) with Eq. (16) then gives an upper bound of m Z /g < 4 TeV (6.5 TeV) at 1 σ (2 σ).

E. Neutrino trident production
The most stringent bound on flavour-diagonal Z couplings to muons arises from neutrino trident production (NTP) ν µ N → ν µ N µ + µ − [15,53]: Seeing as our region of interest is in the small θ R regime, the NTP bound is basically independent of the angle θ R . Taking only the CCFR data [54], we get roughly m Z /g > ∼ 550 GeV at 95% C.L. Compared to τ → µµµ the trident neutrino bound only dominates for very small values of θ R , roughly when θ R < ∼ 10 −3 (see Fig. 2 (right)). For m Z > m Z , the LHC constraints from the process pp → µµZ → 4µ (or 3µ plus missing energy) [55] are currently weaker than NTP [15], but will become competitive with higher luminosities [56][57][58].

F. Phenomenological analysis
Concerning the phenomenological consequences of of our model, let us first consider the implications of h → µτ . In the left plot of Fig. 1 we show the regions in the cos(α − β)-sin(θ R ) plane which can explain h → µτ at the 1 σ and 2 σ level for different values of tan β. Measurements of the h couplings to vector bosons require | cos(α − β)| < ∼ 0.4 [59,60] while the Higgs effects in τ → 3µ and τ → µγ are typically negligible [35]. As a side effect, the h → µτ rate also implies a change in the h → τ τ rate, although this is negligible in regions with small θ R . In addition we show the regions compatible with τ → 3µ for various values of m Z /g . Note that g < ∼ 0.3 in order to avoid a Landau pole below the Planck scale. In summary, small values of θ R can explain the CMS h → µτ excess for moderate to large values of tan β for cos(α − β) 0.1.
In the right plot of Fig. 1 we examine which regions in parameter space can account for B → K * µ + µ − taking into account the constraints from B s -B s mixing. Since  (18)).
we focus on the limit M D → ∞ (i.e. C 9 → 0) we find that unless Γ dL 23 is rather large, B → K * µ + µ − can be explained without violating bounds from B s -B s . Only a very small Γ dL 23 independent region is excluded by NTP. In addition, bounds from τ → 3µ depending on sin(θ R ) can be obtained.
Concerning τ → 3µ, future sensitivities down to Br [τ → 3µ] 10 −9 seem feasible [61] and will cut deep into our parameter space (see Fig. 2). Using the 1 σ limits on h → µτ to fix θ R and B s mixing with C 9 to fix m Z /g -as well as the LHC limit | cos(α − β)| < 0.4we can obtain a lower limit on the rate τ → 3µ which implies tan β > ∼ 18 with current data [50] and tan β > ∼ 61 if branching ratios down to 10 −9 can be probed in the future. This is the main prediction of our simultaneous explanation of h → µτ , B → K * µ + µ − and R(K).
Finally, we remark that a Z-Z mixing angle θ ZZ [45] is induced by the VEV of Ψ 1 [35] which leads to small shifts in the vector couplings of Z to muons and taus and thus ultimately to lepton non-universality [43]. For the values of interest to our study (see Fig. 2), and in the limit m Z m Z , the shift is automatically small enough to satisfy experimental bounds and leads to tiny branching ratios Z → µτ below 10 −8 (for θ R < 0.1). Note that the couplings to electrons and quarks remain unaffected. For m Z m Z , the ρ parameter is enhanced by [45] ρ − 1 1.2 × 10 −4 θ ZZ 10 −3 and is therefore compatible with electroweak precision data (ρ − 1 < 9 × 10 −4 at 2σ [62]) for the parameter space studied in this letter.

IV. CONCLUSIONS
In this letter we showed for the first time that all three LHC anomalies in the flavour sector can be explained within a single well-motivated model: A 2HDM with a gauged L µ − L τ symmetry and effective Z sb couplings induced by heavy vector-like quarks. Except for the τ -µ couplings, the Higgs sector resembles the one of a 2HDM of type-I. Therefore, the constraints from h decays or LHC searches for A 0 → τ + τ − are rather weak and h → µτ can be easily explained in a wide parameter space. The model can also account for the deviations from the SM in B → K * µ + µ − and naturally leads to the right amount of lepton-flavour-universality violating effects in R(K). Due to the small values of the τ -µ mixing angle θ R , sufficient to account for h → µτ , the Z contributions to τ → 3µ are not in conflict with present bounds for large tan β in wide rages of parameter space. Interestingly, B → K * µ + µ − and R(K) combined with B s -B s put a upper limit on m Z /g resulting in a lower limit on τ → 3µ if Br[h → µτ ] = 0: for lower values of tan β the current experimental bounds are reached and future sensitivities will allow for a more detailed exploration of the allowed parameter space. The possible range for the L µ − L τ breaking scale further implies the masses of the Z and the right-handed neutrinos to be at the TeV scale, potentially testable at the LHC with interesting additional consequences for LFV observables. G. D'Ambrosio acknowledges the partial support my MIUR under the project number 2010YJ2NYW. The work of J. Heeck is funded in part by IISN and by Belgian Science Policy (IAP VII/37). We thank Gian Giudice for useful discussions. We are grateful to Wolfgang Altmannshofer and David Straub for useful discussions and additional information concerning the model-independent fit to C µµ 9 and C µµ 9 . Note added : during the publication process of this letter, CMS has released its final analysis of the h → µτ search as a preprint [63], resulting in slightly changed values -Br[h → µτ ] = 0.84 +0. 39 −0.37 % -which have however only a small impact on our study.