Lepton-flavour violating $B$ decays in generic $Z^\prime$ models

LHCb has reported deviations from the Standard Model in $b\to s\mu^+\mu^-$ transitions for which a new neutral gauge boson is a prime candidate for an explanation. As this gauge boson has to couple in a flavour non-universal way to muons and electrons in order to explain $R_K$, it is interesting to examine the possibility that also lepton flavour is violated, especially in the light of the CMS excess in $h\to\tau^\pm\mu^\mp$. In this article, we investigate the perspectives to discover the lepton-flavour violating modes $B\to K^{(*)}\tau^\pm\mu^\mp$, $B_s\to \tau^\pm\mu^\mp$ and $B\to K^{(*)} \mu^\pm e^\mp$, $B_s\to \mu^\pm e^\mp$. For this purpose we consider a simplified model in which new-physics effects originate from an additional neutral gauge boson ($Z^\prime$) with generic couplings to quarks and leptons. The constraints from $\tau\to3\mu$, $\tau\to\mu\nu\bar{\nu}$, $\mu\to e\gamma$, $g_\mu-2$, semi-leptonic $b\to s\mu^+\mu^-$ decays, $B\to K^{(*)}\nu\nu$ and $B_s$--$\overline{B}_s$ mixing are examined. From these decays, we determine upper bounds on the decay rates of lepton flavour violating $B$ decays. $Br(B\to K\nu\nu)$ limits the branching ratios of LFV $B$ decays to be smaller than $8\times 10^{-5} (2\times 10^{-5})$ for vectorial (left-handed) lepton couplings. However, much stronger bounds can be obtained by a combined analysis of $B_s$--$\overline{B}_s$, $\tau\to3\mu$, $\tau\to\mu\nu\bar{\nu}$ and other rare decays. The bounds depend on the amount of fine-tuning among the contributions to $B_s$--$\overline{B}_s$ mixing. Allowing for a fine-tuning at the percent level we find upper bounds of the order of $10^{-6}$ for branching ratios into $\tau\mu$ final states, while $B_s\to \mu^\pm e^\mp$ is strongly suppressed and only $B\to K^{(*)} \mu^\pm e^\mp$ can be experimentally accessible (with a branching ratio of order $10^{-7}$).


I. INTRODUCTION
While most flavour observables agree very well with their Standard-Model (SM) predictions, there are some exceptions in semi-leptonic B decays (see for example [1] for a recent review). LHCb [2] recently found indications for the violation of lepton-flavour universality in the ratio which deviates from the theoretically clean SM prediction R SM K = 1.0003 ± 0.0001 [3] by 2.6 σ. In addition, LHCb has reported deviations from the SM predictions [4][5][6][7] in the decay B → K * µ + µ − (mainly in an angular observable called P 5 [8]) with a significance of about 3 σ [9,10]. Furthermore, also the measurement of Br[B s → φµ + µ − ] disagrees with the SM prediction [11,12] by about 3 σ [6].
Interestingly, three discrepancies can be explained in a model-independent approach by a rather large newphysics (NP) contribution C µµ 9 to the Wilson coefficient of the operator O µµ 9 (the component of the usual SM oparator O 9 that couples to muons, see eq. (5)) [13][14][15][16][17][18][19]. It is encouraging that the value for C µµ 9 required to explain R(K) (with C ee 9 = 0) is of the same order as the one needed for B → K * µ + µ − and B s → φµ + µ − [6,20]. Taking into account the 3 fb −1 data for B → K * µ + µ − recently released by the LHCb collaboration [10], the global significance for a scenario with a non-vanishing NP contribution to C 9 is found to be 3.7 σ (or even 4.3 σ for NP contributing to C µµ 9 only), and 3.13 σ in a scenario with C µµ 9 = −C µµ 10 [18].
Since R(K) is a measure of lepton-flavour universality violation, it has been proposed to search for leptonflavour violating (LFV) B decay modes as well [30]. This is also motivated by the CMS excess in Br[h → µτ ] [31] which can be explained simultaneously together with Br[h → µτ ], R(K), Br[B s → φµ + µ − ] and Br[B → K * µ + µ − ] within a single model [26,27]. While the specific model of Refs. [26,27] predicts only small effects in LFV B decays, the situation could be different in a generic model. In this article we examine the LFV decays B → K ( * ) τ ± µ ∓ , B s → τ ± µ ∓ (and the corresponding µ ± e ∓ channels) studying a simplified model in which the NP effects originate from a heavy new gauge boson Z with generic couplings to quarks and leptons [63]. We thus introduce a Z boson with mass M Z and arbitrary couplings tosb and charged lepton pairs , = τ, µ, e: As the Z is assumed to be much heavier than the scale of electroweak symmetry breaking, its couplings must respect SU (2) L gauge invariance. This implies that the couplings to neutrinos are the same as the ones to lefthanded charged leptons: Γ L i j = Γ L νiνj . To study bounds on the LFV B decay modes, we perform the following steps: FIG. 1: Feynman diagrams illustrating the steps 1-4 of our analysis (see text). The diagrams display the dominant Z constribution to Bs − Bs mixing, 1. From B s -B s mixing we obtain upper limits on Γ L sb as a function of an imposed fine-tuning measure.
4. Taking into account these constraints we derive upper limits on the branching ratios of the LFV de- In Fig. 1 we show the Feynman diagrams for the dominant Z constribution corresponding to the steps 1-4 of our analysis. We apply a similar procedure to µ ± e ∓ final states. In this case with the best bounds on the lepton couplings are coming from µ → eγ and µ → eνν.

II. PROCESSES AND OBSERVABLES
In this section we collect formulae for the LFV B decays and for the processes relevant to constrain these decays, before we study in the next section the phenomenological impact.
A. Bs − Bs mixing B s − B s mixing is governed by the effective Hamiltonian with the full set of ∆F = 2 operators O ( ) i given e.g. in Refs. [34,35]. In a Z model with generic couplings the operators with non-vanishing Wilson coefficients are with O 1 obtained from O 1 by interchanging L ↔ R. The coefficients are given by They enter physical observables, i.e. mass differences and CP asymmetries, via the calculation of matrix elements involving decay constants and bag factors calculated with lattice QCD (see for example [36] for a review). In addition, QCD renormalization group effects must be taken into account. To this end we use the next-to-leading order equations calculated in Refs. [34,35].
b → s + − transitions are governed at leading order in α s by the effective Hamiltonian where again for the primed Wilson coefficients one should replace P L ↔ P R . We have omitted the magnetic operator O 7 contributing to b → s + − transitions through photon exchange. In our model, contributions to C 7 are loop-suppressed, while the coefficients C and C ( ) 10 are induced at tree-level and read As first noted in Ref. [13,37], C µµ

9
< 0 with C µµ 9 , C ( )µµ 10 ∼ 0 gives a good fit to B → K * µ + µ − data. Another interesting solution is given by C µµ 9 = −C µµ 10 [6,18]. As outlined in the introduction, we thus study the two scenarios of vectorial Z couplings Γ L = Γ R ≡ Γ V inducing only a non-vanishing C µµ 9 (possibly accompanied by a small C µµ 9 ), and of left-handed Z couplings inducing C µµ 9 = −C µµ 10 (possibly accompanied by a small C µµ 9 = −C µµ 10 ). In our analysis we use the global fit of Ref. [6,18], resulting for the two scenarios under consideration in at the (1 σ) 2 σ level, respectively. The quoted ranges are in good agreement with preliminary results of Ref. [19].
The Z boson contributes to the process τ → µνν in two different ways. On one hand, it generates loop corrections to the SM diagram of W exchange. This contribution is already present in the lepton-flavour conserving case [25]. On the other hand, it can mediate τ → µνν at tree-level via LFV couplings. In case of the ν τνµ final state, the latter contribution interferes with the SM tree-level amplitude resulting in a decoupling behaviour as 1/m 2 Z , while in the case of other final-state flavours ν iνj it enters the branching fraction only squared, implying a decoupling behaviour as 1/m 4 Z . Including the mentioned contributions, we find for the branching ratio: Note that here we have neglected such Z loop corrections to the SM diagram of W exchange that would generate a neutrino flavour configuration different from ν τνµ as those terms would be both loop-and 1/m 4 Z suppressed. Following Ref. [25] we use the PDG value [38] BR(τ → µν τνµ ) exp = (17.41 ± 0.04)% .
This should be compared to the SM prediction [39] BR(τ → µν τνµ ) SM = τ τ (5.956 ± 0.002) × 10 11 /s . (11) The dominant uncertainty in the SM prediction for the branching ratio comes from the τ lifetime τ τ . Combining the result on the τ lifetime from Belle [40] with previous LEP [41][42][43][44] and CLEO [45] measurements, gives τ τ = (290.29 ± 0.53) × 10 −15 s. Using this value in the SM prediction for Br(τ → µν τνµ ), we find that the experimental value in Eq. (10) is more than 2σ above the SM prediction. Translated into the variable ∆ τ →µνν , one obtains In the case of non-zero values of Γ µe , in which similar contributions to the muon decay µ → eνν are generated, we require This choice restricts corrections to the Fermi-constant, defined through the decay µ → eνν, to the sub per-mille level and avoids in this way conflicts with electroweak precision data.

D. Z couplings to leptons
The Z boson also generates loop corrections to the Z vertex of the SM Z boson, possibly inducing LFV Z couplings. Generalizing the results of Ref. [25] to the case of chiral and LFV couplings, we find for the relevant branching ratios where g L = 1 − 2s 2 W , g R = −2s 2 W (with s W = sin θ W and θ W denoting the weak mixing angle), and Comparing the SM prediction with the experimental results [38] Br one obtains bounds on the Z couplings to muons and taus.

G. Trident Neutrino Production
Bounds on flavour-diagonal Z couplings to muons can also arise from neutrino trident production (NTP), where a muon pair is created by scattering a muon-neutrino with a nucleon: ν µ N → νN µ + µ − [54]. Note that as the flavour of the neutrino in the final state is not detected, one must sum over all three generations in the case of flavour-violating interactions. Generalizing the formula of Ref. [54] to the flavour-violating case, we obtain for the cross section of NTP with However, as we will see in the phenomenology section, even if we use combined bounds from CHARM-II, CCFR and NuTeV, the resulting constraints on the couplings Γ L,R µµ are not very relevant for our analyis. The reason for this is that we are mainly interested in the region of parameter space with small Γ L,R µµ such that Γ L,R τ µ can be sizable without violating the bounds from τ → 3µ (20).

H. Lepton-flavour violating B decays
Here we give formulas for the branching ratios of LFV B decays, taking into account only contributions from the operators O while neglecting contributions from operators with scalar currents not relevant in our model. For B s → + − (with = ) we use the results of Ref. [55] neglecting the mass of the lighter lepton. The branching ratios for B → K ( * ) τ ± µ ∓ , B → K ( * ) µ ± e ∓ are computed using form-factors obtained from lattice QCD in Ref. [56] (see also Refs. [12,57]). The final results read with τ τ = 2 (green), τ → 3µ (red) and aµ (light gray) for m Z = 1 TeV. The 1 σ region allowed from NTP lies between the magenta dashed lines. Although NP effects move aµ to the right direction, it cannot be explained within our model and we do not impose it as a constraint later on in our analysis. Right: Allowed regions in the Γ L µµ − Γ L µτ plane: from τ → µνν for Γ L τ τ = 0 (blue), Γ L τ τ = −2 (yellow), Γ L τ τ = 2 (green), τ → 3µ (red) for m Z = 1 TeV. The contour lines denote the shift in aµ in units of 10 −10 . For regions compatible with τ → µνν, the NP effects in aµ are rather small. Therefore, we do not impose it as a constraint later on. Bounds from NTP lie outside the plotted range and are not shown.
The formula for the branching ratio of B s → + − is symmetric with respect to the exchange of C , while in the case of B → K ( * ) + − this symmetry is broken by lepton-mass effects. There is a small difference between the theoretical prediction for the charged mode B + → K ( * )+ + − and the neutral one B 0 → K ( * )0 + − due to the different B-meson lifetime τ B which we neglected fixing the numerical value of τ B to the one of the neutral meson. Note that the results above are given for − + final states and not for the sum ± ∓ = − + + + − to which the experimental constraints apply [50]:

III. PHENOMENOLOGICAL ANALYSIS
Having identified the processes relevant to constrain our Z model and having specified our treatment of them, we will in this section study their phenomenological impact and quantify the possible size of the LFV B decays.
We start with the lepton sector, where we consider, as mentioned before, the two scenarios with vectorial (scenario 1) and with left-handed couplings (scenario 2). We examine the numerical impact of the leptonic constraints given in Sec. II C-II G. Fig. 2 shows the allowed regions in the plane of the couplings Γ µµ and Γ µτ from τ → µνν for different values of Γ τ τ , as well as the bounds from τ → 3µ and the anomalous magnetic moment of the muon. Note that the experimental value of Br[τ → µνν] is already above the SM prediction by more than 2 σ, and since in addition the tree-level Z contribution interferes destructively with the SM terms, we show the allowed 3 σ regions for this decay. Interestingly, in scenario 1, τ → µνν rules out an explanation of a µ via a non-vanishing Γ V µτ (contrary to claims in Ref. [58] where the constraints from τ → µνν were not considered), while in scenario 2 the interference with the SM terms in a µ is always destructive, even though its contribution is small. The constraints from Z → µ + µ − and Z → τ ± µ ∓ are subleading compared to the bounds from NTP and τ → 3µ for the Z masses under consideration (at the TeV scale or above) and thus not visible in Fig. 2.
In the quark sector we are only interested in the couplings Γ L,R bs mediating b → s transitions. The most stringent constraints on these couplings stem from B s − B s mixing. Using the 95% CL results on ∆m Bs of the UTfit fit of Ref.
collaboration [59][60][61][64] the range 0.90 < R Bs = ∆m Bs ∆m SM Bs < 1.23 (36) is obtained. Denoting R Bs ≡ 1 + ∆R Bs one can derive limits on Γ L sb and Γ R sb from the relation Here, we have assumed that NP effects give small corrections to the SM amplitude of B s − B s mixing, i.e. do not overcompensate it. The coefficients a Bs , b Bs exhibit only a weak logarithmic dependence on M Z (about 3% when varying M Z from 1 to 3 TeV). In the following we neglect this logarithmic dependence and use the values at M Z = 1 TeV (based on formulae from Refs. [34,35]): The bounds resulting from eqs. (36,37) are shown by the blue contour of Fig. 3. As can be seen, the constraints are weakened if Γ L sb and Γ R sb have the same sign with |Γ R sb | |Γ L sb | or |Γ R sb | |Γ L sb |, as a consequence of cancellations in eq. (37). We note that R(K) and B → K * µ + µ − require at the 2 σ level a substantial non-zero contribution to C µµ 9 , eliminating the option |Γ R sb | |Γ L sb |. As an illustration, we show in Fig. 3 the combined constraints from R(K) and B → K * µ + µ − for different values of a vectorial Z coupling Γ V µµ to muons (scenario 1). Note that in principle there is no upper limit on |Γ L sb | as long as R(K) and B → K * µ + µ − allow for small but non-vanishing contributions to the primed operators C 9 and/or C 10 [65]. Therefore, we define the following measure of fine tuning in the B s system quantifying the degree of cancellation encountered in eq. (37). Restricting X Bs to an acceptable value limits the maximal size of the coupling Γ L sb . As we are interested exclusively in scenarios with C µµ 9,10 C µµ 9,10 , we can neglect the (Γ R sb ) 2 term in Eq. (39) and express Γ L sb in terms of the fine-tuning measure X Bs and ∆R Bs as Note that here and in the following we assume that all couplings Γ L,R are real. Using the maximally allowed value for ∆R Bs from eq. (36), we find c Bs = max ∆R Bs /2a Bs ≈ 0.0045 TeV −1 . (41) Inserting the bound on Γ L bs into the formulae for the branching ratios of τ → 3µ and τ → µνν from Sec. II C and II E, we derive upper limits for the coefficient C µτ 9 : Note that in the contraint from τ → µνν we have ne-glected terms decoupling with 1/m 4 Z as they are sub- from the fits of Ref. [6,18].
leading for the range of Z masses we are considering (at the order of TeV or above). For scenario 1 with vectorial lepton couplings we obtain A The bounds from τ → µνν only depend on the finetuning measure X Bs , while the bounds from τ → 3µ also depend on the value of C µµ 9 (and C µµ 10 in scenario 2) determined from the fit to B → K * µ + µ − , B s → φµ + µ − and R(K) data. The latter bounds disappear in the limit C µµ 9 → 0, as in this case the Z µµ couplings may vanish so that the τ → 3µ decay does not receive contributions from Z exchange.
For µe final states we get from µ → eγ and µ → eνν decays (see Sec. II C and II F) where A (1) From the upper bounds on C τ µ 9,10 and C µe 9,10 , we can now determine the maximally allowed branching ratios for the LFV B decays. For B → K ( * ) τ ± µ ∓ and B s → τ ± µ ∓ the maximal values are shown in Fig. 4 for a fine-tuning in B s − B s mixing of X Bs = 20 and X Bs = 100. The kink in the curves occurs at the point where the C µµ 9,10independent constraint from τ → µνν becomes stronger than the constraint from τ → 3µ. Comparing these results to the experimental upper limits in Eq. (32) we see that the current experimental sensitivity is still two orders of magnitude weaker. However, LHCb will be able to achieve significant improvements in these channels.
As the branching ratio for the LFV B decays with µe final states turn out to be quite suppressed, we confine ourselves to displaying the resulting upper limits for scenario 1 (Fig. 5). As we have seen in the case of τ µ final states, the quantitative behavior for scenario 2 is very similar. Due to the stringent bounds from µ → eγ the allowed values are very small and unobservable for the currently favoured C µµ 9 range. The kink is located at the point where the bounds from µ → eγ and µ → eνν are equal. for a fine-tuning of XB s = 100 (solid lines) and XB s = 20 (dashed lines). Note that the limit on Br[Bs → e ± µ ∓ ] is so stringent that it cannot be resolved in the plot. The white area represents the 2 σ-allowed range for C µµ 9 from the fits of Ref. [6,18].

IV. CONCLUSIONS
In this article we have investigated the possible size of the branching ratios of the lepton-flavour violating B decays B s → τ ± µ ∓ , B s → µ ± e ∓ , B → K ( * ) τ ± µ ∓ and B → K ( * ) e ± µ ∓ in generic Z models. To this purpose we have focused on two different scenarios motivated by the model-independent fit to b → s transitions: in scenario 1 we assumed vectorial couplings of the Z to leptons corresponding to NP in the Wilson coefficients C ( ) 9 only (with |C 9 | |C 9 |), whereas in scenario 2 we considered a Z with purely left-handed couplings to leptons corresponding to NP contributions fulfilling C ( ) 9 = −C ( ) 10 (with |C 9,10 | |C 9,10 |). We have found that in both scenarios the branching ratios with µτ final states can be sizable (of the order of 5 × 10 −6 ) if we allow for a significant amount of fine-tuning between the different terms contributing to B s − B s mixing. However, for degrees of fine-tuning below X Bs = 100, the branching ratios for µe final states can only reach ×10 −7 , and this only in a region of parameter space disfavoured by the data on B → K * µ + µ − , B s → φµ + µ − and R(K).
The cancellations in B s − B s mixing that are necessary to permit sizable effects in lepton-flavour violating B decays can only occur if Γ R sb does not vanish but is small compared to Γ L sb (or vice versa) and of the same sign. If the Z couples flavour-diagonal to muons, given the present data on b → sµ + µ − transitions, this implies non-vanishing C µµ 9,10 with |C µµ 9,10 | |C µµ 9,10 |. Future data on b → sµ + µ − transitions constraining the C µµ 9 and C µµ 10 Wilson coefficients can thus rule out the possibility of large lepton-flavour violating B decay rates in such models.