Charged-Lepton Mixing and Lepton Flavor Violation

We present a model for calculating charged-lepton mixing matrices. These matrices are an essential ingredient for predicting lepton flavor-violating rates in the lepton number nonuniversal models recently proposed to explain anomalies in B-meson decays. The model is based on work on"constrained flavor breaking"by Appelquist, Bai and Piai relating the charged-lepton mass matrix, M_l, to those for the up and down-type quarks, M_{u,d}. We use our recent model of lepton nonuniversality to illustrate the magnitudes of flavor-violating B-decay rates that might be expected. Decays with mu tau final states generally have the highest rates by far.

These three measurements were made in the low q 2 = M 2 region of 1.0-6.0 GeV 2 , away from charmonium resonances in the + − spectrum.
• The joint CMS-LHCb measurement [13] B(B s → µ + µ − ) exp = (2.8 +0.7 −0.6 ) × 10 −9 = (0.76 +0.20 −0. 18 Although this is consistent with the SM prediction [14], the central value is about 25% low -as it is for R K [15]. The R K -measurement suggests lepton nonuniversality (LNU) occurs in b → s + − transitions; the other measurements are consistent in magnitude and sign. It is no wonder, then, that they have inspired a number of LNU models of new physics (NP) above the electroweak energy scale, involving the exchange of multi-TeV particles [16,17,18,19,20,21,22,23,24,25,26,27,15,28,29,30,31,32,33,34,35,36,37,38,39,40]. LNU interactions at high energy are accompanied by LFV interactions unless the leptons involved are chosen to be mass eigenstates [15]. Such a choice is an act of fine tuning in the absence of a dynamical or symmetry mechanism justifying it. 1 Further, since charged leptons (and quarks) are massless at Λ LN U , far above the weak scale, it is difficult to understand the motivation or need for flavor-invariant Yukawa couplings there. If the anomalies reported by LHCb hold up, LFV decays such as B → K ( * ) µe and B → K ( * ) µτ should occur at rates much larger than in the SM due to tiny neutrino masses alone. The purpose of this paper is to present a model for estimating these and other LFV rates implied by new LNU interactions.
LHCb data suggest that LNU is affecting muons but not electrons. To describe this, a simple model was adopted in Ref. [15] (hereafter referred to as GGL) in which a heavy Z boson couples only to third-generation quarks and leptons, namely, This chiral structure is consistent with B-decay data which is well fit if the SM and NP contributions to the b → s + − interaction are a product of left-handed currents (LL) [22,23,24,41]. In Eq. (4), The primed fields refer to the gauge basis, the one in which the charged weak currents are generationuniversal. 2 They are related to mass-eigenstate (unprimed) fields by unitary matrices U d L and U L : The interaction responsible for the discrepancies in R K , B + → K + µ + µ − , B s → µ + µ − and P 5 is then In GGL we said that the hierarchy of the Cabibbo-Kobayashi-Maskawa (CKM) matrix V CKM = U u † L U d L for quarks and the apparent preference of the new physics for muons over electrons suggest that |U d, Then, in order that this Hamiltonian deplete the SM contribution, we assumed GU d * L33 U d L32 < 0. This sign choice is correct if U d L V CKM . In truth, however, we know little about U d L and U L other than V CKM = U u † L U d L and that U L plays a similar part in the less well-measured PMNS matrix, V P M N S [43]. Experimentalists need better targets. A recent paper by Boucenna, Valle and Vicente [35] made a first stab at this by assuming U L = V P M N S . In our opinion, V P M N S = U ν † L U L seems likely to be strongly influenced by the unknown neutrino mixing matrix. Ref. [34] discussed lepton flavor mixing in the context of LNU due to a leptoquark interaction [22]. While we will present results for the Z -induced H N P , our scheme for charged-lepton mixing is independent of the dynamical nature of LNU and LFV.
The model we propose for calculating charged-lepton mixing matrices, U L,R , is based on a recent paper on "constrained flavor breaking" by Appelquist, Bai and Piai (ABP) [44]. They assumed a global constrained flavor symmetry group, SU (3) 3 , broken by just two Yukawa spurions. This implies one equation among the three Yukawa matrices, Y u , Y d , Y . They are related to the quark and charged-lepton mass matrices M a in where v is the Higgs vacuum expectation value. The matrices U a L,R bring these to real, diagonal, positive form: The ABP model is based on the moose diagram of Fig. 1. Requiring (a) that the quark doublet and singlet fields, Q L , u R and d R , must be assigned to different SU (3)'s (to have realistic masses and V CKM ); (b) that L L and e R likewise be assigned to different SU (3)'s; and (c) that L L and e R be assigned to SU (3) groups other than Q L 's (to avoid M ∝ M u or M d ), leaves six possibilities [44]. The one chosen by ABP is depicted Fig. 1. Having taken Y u and Y independent, Y d is predicted up to a constant, η: As ABP were interested only in masses of the charged leptons, not their mixing, they did not consider assignments differing by an interchange of L L and e R which just interchanges Y and Y † . But, this swaps M and M † and, hence, U L with U R , resulting in different mixing factors for LNU/LFV processes. ABP rejected choosing Y u and Y d as the flavor-breaking spurions because that implies Y = ηY † d Y u and, hence, unrealistic charged-lepton masses. We agree. They also rejected the possibility Y u = ηY d Y , arguing that it has difficulty obtaining a large enough top Yukawa coupling. But η is a free parameter of unknown origin, and there is considerable freedom in choosing the textures for the Y a , so we will consider this case.
We will see it is closely related to the case ABP considered. Thus, we consider four cases. Written as a relation from which M is determined, they are (ignoring the dimensionful η-factor): By a special choice of quark bases, neither the gauge nor mass bases defined above, ABP obtained the mass matrix and, hence, mass ratios for the charged leptons from Eq. (9) in terms of the known quark masses and CKM matrix elements [43]. No assumption of particular quark mass textures was required. While their results are not in agreement with data, they are not all that bad, so there is promise in their approach. But, the matrices diagonalizing their lepton mass matrix are not transformations from the gauge to the mass basis, and thus cannot be used for turning H N P in Eq. (4) into predictions of LFV rates. For that, we need specific textures for M u,d in the gauge basis, ones that provide a reasonable account of the quark masses and V CKM .
Fortunately, quark mass textures good enough for our purpose exist; see, e.g., Ref. [45]. We use ones developed in connection with a scenario for solving the strong-CP problem in QCD [46]. In this scenario, the phases in M u,d are rational multiples of π so that they easily satisfyθ The The notation is (|M q, ij |, arg(M q, ij )). The motivation for these mass textures is explained in Appendix B of Ref. [46]. Note that, since quark masses are multiplicatively and universally renormalized above M t , the lepton mass textures in Eq. (10) are insensitive to QCD running from M t to Λ N P . The CKM matrix obtained by diagonalizing M u,d , removing its unphysical phases and casting it in standard form [47], is This reproduces measured CKM matrix entries to within a few per cent, except for V ub and V td which are within 20% [48]. The U L,R are obtained (up to a diagonal matrix of pure phases) by diagonalizing For cases A and C (and for cases B and D) Therefore, the (dimensionless) eigenvalues of M 2 , LL (C) are the inverses of those of M 2 , LL (A), i.e., and U L,R (C) are the same as U L,R (A) with their first and third columns interchanged. The mass-squared matrices for case A are The predicted and measured ratios of the lepton masses are The predicted ratios are not great, but they do exhibit a qualitatively correct hierarchy. Different quark mass textures will lead to different ratios. 4 The lepton mixing matrices for this case are These mixing matrices, or ones developed from other quark-mass textures and the ABP ansatz, can be used to predict LNU and LFV rates in any NP model of these processes. For our H N P , Eq. (4), the elements of interest in U L are the third row, v τ L . In particular, the amplitudes for 31 |, contrary to our naive expectation that these matrices have a CKM-like hierarchy [15]. Even more surprising |U R,31 | |U R,32 | |U R,33 |. Note that this means that the U L,3i of case D are CKM-like, as naively expected. These features are a consequence of the block-diagonal M 2 ,LL and M 2 ,RR , the latter exhibiting an extreme example of levelcrossing. In turn, these trace back to the textures of M u and M d (and the ABP ansatz). M u is (2 × 2) ⊕ (1 × 1) block-diagonal and employs a see-saw to make m u m c without an O(m u ) matrix element. Approximately the same structure in M d plus U u L 1 lead to the famous relation tan θ C ∼ = θ 12 m d /m s in V CKM . Finally, we apply these results to our model "third-generation" Hamiltonian, Eq. (4), and evaluate branching ratios for the LFV processes B → K ( * ) ± i ∓ j and B s → + i − j . Since these rates will be proportional to |U * L3i U L3j | 2 , both lepton charge assignments may be combined. The first order of business is to note that cases B and C are excluded in our model. In those cases, |U L31 | 2 is not much smaller than |U L32 | 2 , implying R K > ∼ 1.
Here [15], where V = V CKM , C e 9 is the Wilson coefficient for the operator V * tb V ts in the quark-mass model of Ref. [46], and p is the momentum of the outgoing lepton in the B s rest frame. The value of ρ N P is obtained from the global-fit result C 9,N P −12% C 9,SM [24], rather than from R K alone. This ρ N P applies to axial-vector amplitudes as well because the SM interaction renormalized at m b is pure LL to a good approximation.
From Eq. (25) and the calculated U d, L matrices, one can estimate the G-coupling strength. For cases A and D one has G 4.3 × 10 −8 GeV −2 and 1.8 × 10 −6 GeV −2 . These imply the approximate upper bounds Λ N P = 1/ √ G = 4.8 TeV and 745 GeV, respectively. These mass scales seem low for a Z , but it must be remembered that it couples primarily to the third generation.
There are two approximations in Eq. (23) as applied to B → K + − ratios. The denominator is best-measured for B + → K + µ + µ − ; it is B(B + → K + µ + µ − ) = (4.29 ± 0.22) × 10 −7 integrated over the full q 2 -range, 0-22 GeV 2 [5]. This integration extrapolates over most of the charmonium region, q 2 = 8-15 GeV 2 , ignoring the presence of the narrow resonances. Charmonium resonances do not, of course, influence the numerators, but we do not know whether LFV searches will include this region. Second, and potentially more important, LFV modes to τ 's have smaller phase space than those with µ + µ − . For these semileptonic decays, this effect is accounted for using the results of Ref. [36]. Employing our own calculations, we have corrected for phase space the B s → decay rates involving τ 's. Our results are shown in Table 1. As was to be expected for our third-generation H N P , modes involving µτ have the largest rates followed by eτ with rates smaller by one or more orders of magnitude. Rates for the experimentally easier eµ modes are very small and may be beyond reach in the near future; the one exception in our model is B + → K + eµ in case A. The large B s → µτ rate predictions are not yet excluded. The best public limits on these LFV modes are also listed in Table 1 [49,43].
In conclusion, it is natural to ask how general are our results; are they to be expected in other NP models of the B-decay anomalies or in other schemes for calculating the mixing matrices? Particularly, is the relative importance of B → Xµτ that we found likely to be a common feature of such models? It is hard to be sure, of course, but we do believe it is. As we emphasized of our H N P , the LHCb data strongly points to the third generation, or at most just the second and third generations, as the seat of lepton nonuniversality. Further, the hierarchy of charged-lepton masses -not unlike that for the quarks -suggests blockdiagonal mass matrices and, therefore, mixing matrices somewhere along the line from our original CKM-like expectation to the ones we found in Eqs. (21,22) from the ABP ansatz.
These expectations can be compared with those obtained within other proposed flavor models. Our prediction of a generic enhancement over the SM rate of decay modes involving the third generation is also advertised in the class of models discussed in Ref. [38], although they have unobservable LFV by construction. The only model allowing for a direct comparison is Ref. [35]. For the B → K transitions to either eτ or eµ, the branching-ratio ranges predicted in our cases A and D encompass those predicted in their model. In the µτ case our predictions are above theirs in both A and D cases, although they also predict a relative enhancement of this channel with respect to the other LFV modes.
Therefore, in addition to more firmly establishing the apparent lepton nonuniversality in B decays, it is important that LHCb and other experiments mount searches for lepton flavor violation, with special attention to improving significantly the limits on µτ and even eτ decay modes.