Lepton Flavor Non-Universality in B-meson Decays from a U(2) Flavor Model

We address the recent anomalies in semi-leptonic $B$-meson decays using a model of fermion masses based on the $U(2)$ flavor symmetry. The new contributions to $b \to s \ell \ell$ transitions arise due to a tree-level exchange of a $Z^\prime$ vector boson gauging a $U(1)$ subgroup of the flavor symmetry. They are controlled by a single parameter and are approximately aligned to the Standard Model prediction, with constructive interference in the $e$-channel and destructive interference in the $\mu$-channel. The current experimental data on semi-leptonic $B$-meson decays can be very well reproduced without violating existing constraints from flavor violation in the quark and lepton sectors. Our model will be tested by new measurements of $b \to s \ell \ell$ transitions and also by future electroweak precision tests, direct $Z^\prime$ searches, and $\mu$-$e$ conversion in nuclei.


Introduction
While direct searches for new particles at the LHC have so far been inconclusive, recent results from the LHCb collaboration on semi-leptonic B-meson decays [1][2][3] might provide the first indirect hint of new physics beyond the Standard Model (SM).Starting with the 3σ anomaly in B → K * µ + µ − angular observables [1], several other observables involving b → s transitions have produced significant deviations from the SM predictions.The most notable is the ratio of B ± → K ± µ + µ − to B ± → K ± e + e − branching ratios measured as R K = 0.745 +0.097  −0.082 [2] (deviating from the SM prediction by 2.6σ), as in this case the SM prediction R K ≈ 1 can be calculated with a very good accuracy.The measured branching fraction of B s → φµ + µ − [3] is also low compared to the SM prediction.
The purpose of this work is to provide a predictive model of this kind.We address the anomalies in b → s transitions in the context of a light Z vector boson, whose couplings to fermions are governed by an underlying U (2) F symmetry that explains fermion masses and mixings.The original U (2) models proposed in the context of supersymmetry [39,40] have been disfavored by precision measurements in the B-factories [41], as they predicted the relation V ub /V cb = m u /m c which was not borne out experimentally.However, it is not difficult to modify this prediction with a more general U (1) F charge assignment, as demonstrated in Ref. [42].
The model that we are presenting here is essentially a non-supersymmetric version of the one in Ref. [42], in which the dominant source of deviations from the SM is due to the tree-level exchange of the Z gauge boson associated to the U (1) F flavor group.Similarly to the supersymmetric model, the couplings of the Z are approximately U (2) F symmetric, and flavor violating effects in the quark sector are suppressed by the small CKM mixing angles involving the 3rd generation.In contrast to Ref. [42], we do not demand that the U (1) F charges are compatible with SU (5) grand unification.This generalization gives us more freedom in the charged lepton sector to address the observed anomalies in b → s transitions.Once this is achieved, the parametric freedom in the model is to a large extent fixed by matching to the observed quark and charged lepton masses and quark mixing angles.
In our scenario, the deviations in b → s observables arise from a simple pattern of Z contributions to the 4 relevant Wilson coefficients C ee,µµ 9,10 .Namely, the new contributions are aligned with the SM one (i.e.approximately left-handed) and controlled by a single parameter (the ratio of the U (1) F gauge coupling and the Z mass multiplied by the b L -d L mixing angle) that sets its overall magnitude.Moreover, they interfere constructively in the electron channel and destructively in the muon channel.As a consequence, we predict a simple pattern for the relevant b → s amplitudes: in the electron channels the SM predictions are rescaled by a factor r e > 1, whereas in the muon channels they are rescaled by a correlated factor r µ < 1.The current experimental data on b → s transitions determine the overall normalization of the Z contribution.This in turn fixes the predictions for other flavor-violating observables up to O(1) coefficients that span the parameter space of our model.Comparing that with existing constraints from ∆F = 2 and LFV observables, we obtain bounds on these O(1) coefficients.The strongest ones come from B s and kaon mixing, electroweak precision measurement in LEP-2, and, especially, from µ-e conversion in nuclei.These bounds disfavor large regions of the parameter space, but they nevertheless leave enough room to address the B-meson anomalies.The corollary is that our scenario will be decisively tested not only by upcoming new data from LHCb, but also from near future tests of LFV in µ → 3e decays and µ-e conversion in nuclei.Last but not least, if the Z boson couples to fermions with electroweak strength, it is within the kinematical reach of LHC.
This paper is organized as follows.In Section 2 we define the setup of the U (2) F flavor model and use its predictions for fermion masses and mixings to determine the couplings of the Z gauge boson to fermions.In Section 3 we demonstrate that the resulting contributions from tree-level Z exchange to Wilson coefficients controlling b → s transitions allow one to address the B-meson anomalies.In Section 4 we study other constraints on the parameter space, and show that electroweak precision tests in LEP-2 and µ-e conversion in nuclei provide important constraints.We conclude in Section 5.In Appendix A we provide analytical results for the eigenvalues and mixing angles of the quark Yukawa matrices.

The Model
In this section we define our model with a U (2) F flavor symmetry.We first study its predictions concerning the fermion masses and mixings and demonstrate that the observed patterns in the quark and lepton sector can be reproduced.Then we discuss the physics of the Z boson associated to the U (1) F factor of the flavor group.This degree of freedom will be the origin of lepton flavor violation in the B-meson sector that we discuss in the next section.

Flavor Symmetries
We first consider an extension of the SM with the global symmetry U (2) F ≡ SU (2) F × U (1) F acting in the fermion's generation space.Here we restrict to the effective description involving only SM fields and spurions parametrizing the breaking of SU (2) F × U (1) F .We assume that the additional degrees of freedom needed to UV-complete this theory are heavy enough not to play role in the low-energy dynamics, i.e. the cutoff-scale Λ of the effective theory is in the multi-TeV range.The first two generations transform as a doublet under SU (2) F , and the third generation is an SU (2) F singlet.The U (1) F charges of all fermions are treated as free parameters for a while; they will be fixed later to reproduce the observed mass and mixing hierarchies.The Higgs field is a total flavor singlet.The breaking of the flavor symmetry is described by two scalar spurions: φ transforming as 2 X φ , and χ transforming as 1 −1 .These fields acquire the following vacuum expectation values (VEVs): where we assume φ,χ 1.We also define φ ≡ iσ 2 φ * which transforms as 2 −X φ .In Table 1 we list the field content and their general transformation properties under the flavor group.In the next sections we will specify the U (1) F charges X F i needed to reproduce fermion masses and mixings.As the fermions are in general charged under U (2) F , Yukawa couplings require additional spurion insertions in order to be U (2) F -invariant.This leads to non-renormalizable interaction suppressed by appropriate powers of Λ.After inserting the spurion VEVs the cutoff dependence drops out, and Yukawa hierarchies arise from powers of the small parameters φ,χ .The resulting Yukawa matrices are of the form where D = Q, L, and S = U, D, E. In each entry we omitted terms suppressed by more powers of φ,χ coming from higher-dimensional terms in the effective theory.The absolute value appears because only positive powers of χ or χ * are allowed in the effective Lagrangian.Note that, in contrast to the supersymmetric U (2) model in Ref. [42], there are no holomorphy constraints, which leads to a more general Yukawa pattern.
We move to discussing the consequences of the Yukawa pattern in Eq. (2.2) for the fermion masses and mixing.

Quark Masses and Mixings
In the quark sector, we fix X Q 3 = X U 3 = 0, so that the top Yukawa coupling is not suppressed.Furthermore, we impose the following constraints on the charges: With these constraints, we find the following up-and down-quark Yukawa matrices: where we have defined The y 11 , y 13 , and y 31 entries are not exactly zero but one can show they yield subleading corrections to quark masses and mixings relatively suppressed at least by 2 φ .Thus, effectively, three texture zeros appear in the Yukawa matrix, much as in the supersymmetric models [42].It was pointed out long ago [39] that the presence of these three texture zeros leads to relations among quark masses and mixings that work remarkably well from the phenomenological point of view.
Yukawa matrices of the form in Eq. ( 2.4) can be diagonalized in a fully analytic way.However, it is more convenient to first illustrate the most important points in perturbative analysis.Indeed, since mixing angles in the left-handed (LH) quark sector are known to be small, one can use them as the small parameter in which the eigenvalues and the remaining mixing angles are expanded.Ignoring O(1) coefficients, this gives the following rough estimates for the eigenvalues and the CKM matrix: We thus have 5 small parameters that set the order of magnitude of 8 observable quantities.Expressed in powers of the Cabibbo angle λ ≈ 0.2, the magnitudes of the parameters consistent with experiment is where the y s /y c hierarchy must be explained by order 1 factors.The parametric size of rotation angles and matrices (see Appendix for our conventions) is then given by and The constraints in Eq. (2.7) are recovered (up to a mismatch in u 12 that again is ascribed to order one factors) when the U (1) F charges are fixed as and the spurion VEVs are of the order χ φ ∼ λ 2 .One robust conclusion is that, given the observed masses and mixings, the U (1) F charges in the RH down sector should be universal.This has important consequences for phenomenology, as we will discuss later on.
We now improve on the above rough estimates, taking into account the O(1) coefficients.For our purpose, it is convenient to express the observables in terms of physical quark masses and the unitary rotations that connect the flavor and mass basis.To this end we pick 4 rotation angles: s Lu 23 , s Ru 23 , s Ld 23 , s Rd 23 .As we show in Appendix, the remaining rotation angles, up to percent corrections, can be expressed in terms of these 4 angles and the quark mass ratios:

Charged Lepton Masses
In the lepton sector we focus on the charged lepton masses, and we ignore here the neutrino masses 1 .Therefore the rotation angles in the charged lepton sector are not constrained by phenomenology, which leaves more freedom in the choice of the model parameters.The simplest possibility is to take the U (1) F lepton charges to be compatible with SU(5) grand unification, that is to say, the same as for the down-type quarks [42].However, one can show that such a choice does not allow to address the lepton non-universality in b → s transitions, which is the primary goal in this paper.Therefore we make a different choice of the U (1) F charges: where X E 1 does not enter into the lepton mass matrix and is left unspecified for the moment.This choice leads to the following lepton Yukawa matrix: 31 that set the magnitude of the 1-2 and 1-3 mixing angles.We will use this freedom later when addressing the b → s anomalies in a way that avoids phenomenological constraints.

Z -Boson
We now extend the model by promoting U (1) F to a local symmetry (as in Ref. [42]).We assume that the associated gauge boson is relatively light, with a mass in the TeV range.
Note that the U (1) F symmetry without additional fermions is necessarily anomalous if the model explains fermion mass hierarchies.This is due to the relation [44][45][46], where C 3 is the anomaly coefficient of the mixed SU (3) 2 U (1) F anomaly.As U (1) F is spontaneously broken by the VEVs of φ and χ at a scale v ∼ Λ, we assume that the anomaly is cancelled by unspecified dynamics (involving new chiral fermions) at the scale Λ U V 4πv .Since the new gauge boson has a mass given by M Z = g v , it can easily be the lightest new degree of freedom when g is sufficiently small.We therefore ignore the additional heavy dynamics and concentrate on the effects of the Z gauge boson.
In the flavor basis, Z couples to each fermion proportionally to its We have fixed these charges (except for X E 1 ) to fit the observed fermion mass hierarchies.That fit also determines the unitary rotations that connect the flavor and the mass basis.Therefore, flavor non-universal effects mediated by Z are predicted in our model, up to an overall normalization determined by the Z mass and gauge coupling, and up to the freedom of choosing X E 1 and order one Yukawa factors.In particular, the SU (2) F structure for the first two generations implies that flavor changing effects are entirely determined by the 3rd row of the rotation matrices.In the mass basis, the Z couplings take the form ∆ ∆ Note that flavor-violating couplings are proportional to the charge difference X 1 − X 3 .As a consequence, with the charge assignments in Eq. (2.10) there is no flavor violation in the RH down sector: ∆ For the LH down quarks we find Since s Ld s Ld 23 (see Eq. (2.11)), the largest flavor violating effect of Z is in b → s quark transitions.This will be handy for addressing the recent B-meson anomalies, as we will discuss in the next section.For LH and RH up quarks we find ∆ where X = {L, R}.
In the charged lepton sector the Z couplings depend on the charge X E 1 and to good approximation only on the 1-3 rotation angles. (2.27) The diagonal muon Z coupling is different from the electron and tau one at leading order, which is due to the 2-3 inversion in the LH sector.This feature of our model will allow us later to address the anomalies in B-meson decays involving muons and electrons.In the RH sector electrons and muons have approximately the same coupling to Z and the dominant flavor non-universal effects must involve the tau lepton.The largest flavor violating effects occur in the LH µ-e and RH τ -e transitions.Note that the rotation angles setting the magnitude of these lepton-flavor-violating effects are fixed up to O(1) factors and expected to be tiny, see Eq. (2.19).As a result, lepton flavor violation in our model is suppressed at least by a factor of order 2 χ / φ ≈ 5 × 10 −4 as compared to violation of lepton flavor universality.

Phenomenology of b → s Transitions
We now turn to the predictions for b → s transitions.Our main goal is to address the recently observed violation of lepton flavor universality in B-meson decays [2].In our model, this anomaly is due to the exchange of a U (1) F Z boson with mass in the multi-TeV range.Low-energy observables are controlled by the 4-fermion effective operators that arise from integrating out the Z at tree level, The 4-fermion operators in Eq. (3.1) include the ones relevant for B → K decays which are customarily parametrized by the following effective Hamiltonian (see e.g.Ref. [21]): Note that the analogous 4-fermion operators with RH quarks are not generated in our model ( C 9 = C 10 = 0), as a consequence of the universal U (1) F charge assignment in the RH down sector.Matching Eq. (3.1) and Eq.(3.2), the Wilson coefficients are given by Focusing for now on the lepton flavor conserving operators with electrons or muons, we have where we defined the parameter k as The explicit expressions for the Wilson coefficients in Eq. (3.4) imply that large corrections to B-meson decays involving muons are correlated with comparable corrections to the analogous observables involving electrons.While new physics in the muonic sector alone gives the most economical explanation of the LHCb anomalies (including R K [12,14]), it has been emphasized that large corrections in electron channels are not only allowed but could also (slightly) improve the goodness of the global fit [13].
We can now identify the parameter space of our model where the measurements of semileptonic b → s transitions with = e, µ are best reproduced.As can be seen from Eq. (3.4), these observables depend on just 2 parameters: k defined in Eq. (3.5), and the U (1) F charge X E 1 .For the moment, we treat them as free parameters, various precision constraints will be discussed in the next section.In order to find the best fit region for k and X E 1 , we use the result of Ref. [13].The authors provide the results of the fit in the 2D planes (C ee 9 , C µµ 9 ), (C ee 10 , C µµ 10 ), (C µµ 9 , C µµ 10 ) and (C ee 9 , C ee 10 ).Ignoring possible correlations in the full 4D likelihood, we identify the allowed range for k for discrete values of X E 1 by requiring to simultaneously remain inside the 68% or 95% confidence level regions in every 2D plane.We obtain: This simplified analysis suggests that, while a reasonable fit to the b → s data is possible for a range of X E 1 , the best case scenario is X E 1 = 0.In the rest of this paper we focus on that particular choice.In this case, the new physics contributions mediated by Z are purely left-handed, C ee 9 = −C ee 10 , C µµ 9 = −C µµ 10 , and we can derive constraints on k using the 2-parameter fit of Ref. [13] for precisely this case.This way, we find that the 1σ confidence interval is k ∈ [1.9, 4.9] and the 2σ one is k ∈ [0.2, 6.5].The allowed region of the parameters C ee 9 -C µµ 9 parameter space overlaid with the prediction of our model is displayed in Fig. 1.We turn to discussing predictions of our model.The case with X E 1 = 0 is particularly simple because the SM contributions to the effective Hamiltonian in Eq. (3.2) are also purely left-handed, C SM Figure 1: Global fit results from [13], the blue (light blue) domain corresponds to the 1σ (2σ) region.The red line is our model prediction for X E 1 = 0 and varying k.
For k ∈ [1.9, 4.9] we thus predict an enhancement in all the electron channels by r e ∈ [1.17, 1.48], and a suppression in all the muon channels by r µ ∈ [0.73, 0.89].As a reference, in Table 2 we show the measured values and the SM predictions for various b → s observables.It is remarkable that most observables in the muon channel shows a deficit compared to the SM predictions, while the ones in the electron channels show some (albeit not statistically significant) enhancement.Lepton flavor universality is often tested by measuring ratios of branching fractions of semileptonic B-meson decays.In our model we have where X = K, K * , φ, X s .The interval k ∈ [1.9, 4.9] corresponds R X ∈ [0.50, 0.76], which should be compared to the LHCb measurement R K = 0.745 +0.097 −0.082 .Future improvements in the precision of R K and other measurements will be crucial for testing our model, since we predict a rather low value for these observable.This is actually supported by measurements of inclusive B → X s decay ratios from BaBar (R Xs = 0.58 ± 0.19) [47] and Belle (R Xs = 0.42±0.25)[51], although with large errors.Another test of lepton-non-universality is provided by double ratios [52] such as R K * /R K .In our model, as in any scenario with C9,10 = 0, all these double ratios are predicted to be equal to one.
Finally, we comment on the predictions concerning LFV B-meson decays.The Wilson coefficients of 4-fermion operators mediating these decays are suppressed by additional powers of small parameters, e.g.

Constraints from ∆F = 2 Observables
The Z boson exchange generates 4-quark operators mediating ∆F = 2 transitions.In the notation of e.g.Ref. [55] these are denoted O 1 (with 4 LH quarks), Õ1 (with 4 RH quarks), and O 5 (with 2 LH and 2 RH quarks).Since ∆ = 0, for the down-type quark only O 1 is generated.Their (in general complex) Wilson coefficients are given by and numerically one has (using m d /m s ≈ 0.05) where k is defined in Eq. (3.5) and it needs to be O(1) for the model to address the Bmeson anomalies.We have also approximated s Ld 13 ≈ −s Ld 23 s Ld 12 , which slightly overestimates the Wilson coefficients, see Eq. (2.11).These expressions have to be compared to the bounds from K-mixing taken from Ref. [55] (Im) and Ref. [56] (Re), and the bounds from B-mixing taken from Ref. [57]: This shows that for s Ld 23 ∼ V cb and k in the experimentally preferred range k ∈ [1.9, 4.9] the bounds from K, B and B s mixing are satisfied, even for an O(1) phase in Im C 1 K .Turning to the up sector, 4-fermion operators with both LH and RH fermions are generated, as a result of a non-universal U (1) F charge assignment.In particular, for the ∆C = 2 operators we have (with m u /m c ≈ 0.002)  there are no further bounds on our model from D−mixing.
and for the s → d transitions by where we approximated ∆ bd L ≈ −s Ld  4, adapted from the case of composite leptoquarks [28].From this table, it is easy to verify that for the experimentally preferred range k ∈ [1.9, 4.9] all the bounds are satisfied.

Lepton flavor violation
From Eq. (2.27), the largest LFV Z couplings are the ones to LH muons and electrons.These are constrained by several precise measurements of LFV µ → e transitions.First, we have the µ → 3e decay with the branching fraction where v = 246 GeV.This should be compared with the experimental limit from Ref. [59]: This limit can be violated for larger values of the parameter k.In particular, for X E 1 = 0 and k in the experimentally favored range k ∈ [1.9, 4.9] we get the constraint on the mixing angles: Formally, s Le 13 is a free parameter, therefore Eq. (4.15) can always be satisfied with an appropriate choice of the lepton Yukawa couplings.However, our philosophy is to explain the flavor hierarchies with all Yukawa couplings in Eq. (2.2) being O(1), in which case the natural value is s Le 13 ∼ 10 −4 .In this respect, Eq. (4.15) forces us into a less natural corner of the parameter space and suggest a value of k close to the lower 1σ boundary.We note that the experimental sensitivity to the µ-e conversion rate is expected to improve by many orders of magnitude in the near future [65][66][67].In case of a null result, our model will no longer be an attractive solution to the B-meson anomalies.

Electroweak precision tests
Integrating out Z induces lepton-number conserving 4-fermion operators which can be constrained by electroweak precision tests.Here, we focus on the 4-lepton operators which give the strongest bounds due to large U (1) F charges of leptons.At leading order, these do not affect Z-pole observables measured in LEP-1 and SLC, but they can be constrained by off-Z-pole fermion scattering in LEP-2.We parametrize these operators as For X E 1 = 0 the non-zero Wilson coefficients are Note that the sign of each contribution is fixed, in particular the contribution to [c LL ] ee is always negative in our model.We calculated the impact of these operators on the LEP-2 observables quoted in Ref. [68].We used the total cross section and asymmetries of e + e − → µ + µ − , τ We also comment on the corrections to lepton flavor conserving muon decays.Loops with a Z boson result in the following 1-loop correction to the µ → eν µ ν e decay width [21]: where

.21)
The muon lifetime measurement does not constrain new physics by itself, because it is used to extract the SM input parameter G F (equivalently, the Higgs VEV v).However, indirectly, new physics contributions to G F shift other observables (for example m W , Z-pole asymmetries, etc.) away from the SM predictions.To estimate the resulting constraints, we note that the effect in Eq.M Z' @TeVD g' Figure 2: For X E 1 = 0, the region of the M Z -g plane of our model excluded by resonance searches at the LHC (red).We also show the indirect constraints from the 2-fermion production in LEP-2 (black mesh).The green regions correspond to s Ld 23 = 2|V cb | and the parameter k in Eq. (3.5) in the range favored by the B-meson anomalies at 1 σ k ∈ (1.9, 4.9) (darker) and at 2 σ k ∈ (0, 2, 6.5) (lighter).

Z searches in colliders
Finally, the parameter space of our model is constrained by direct searches for resonances in colliders.Since addressing the B-meson anomalies requires M Z /g ∼ 20 TeV, the Z boson predicted by our model is within the kinematic reach of LHC for g of electroweak strength or smaller.Note that the direct searches probe separately the Z mass and coupling constant, unlike all previously discussed observables that depended on these parameters only via the combination M Z /g .Given the charge assignments in Eq. (2.10) and Eq.(2.16), the branching fraction of Z into dilepton final states is significant.In particular, for X E 1 = 0, we have Br(Z → ee) ≈ 14%, Br(Z → µµ) ≈ 6%, (4.22) and the strongest constraints are expected from the di-electron channel.In Fig. 2 we plot the constraints in the M Z -g plane based on the CMS search for di-electron resonances in the LHC at √ s = 8 TeV [70].These constraints imply M Z 3 TeV and g 0.1 in the region of the parameter space favored by the B-meson anomalies.Note that the direct limits are complementary to the indirect ones from LEP-2.The latter would allow us to address the B-meson anomalies with a light (m Z 2 TeV) and very weakly coupled Z ; such possibility is however excluded by the resonance searches.will be provided by further of lepton flavor universality in LHCb and B-factories, as well as by future experiments looking for µ − e conversion in nuclei.Apart from indirect searches, the Z boson is likely within the reach of the LHC run-2, and should first show up in the di-electron channel, as a result of its large coupling to electrons.

9 ≈ −C SM 10 ,
with C SM 9 (m b ) ≈ 4.2.Therefore the new physics contributions interfere constructively with the SM in the e-channel and destructively in the µ-channel, resulting in a simple rescaling of B-meson decay rates by the factors r e and r µ that are the same for all b → see and b → sµµ processes 2 ,

4. 2
Semileptonic decays in b → d and s → d transitions We now turn to semileptonic decays involving b → d and s → d transitions with electrons or muons in the final state.The relevant Wilson coefficients for the associated 4-fermion operators for b → d transitions are given by

23 m d /m s and ∆ sd L ≈ s Ld 23 2 m
d /m s .The upper bounds are summarized in the Table

( 4 . 2 ¯ 1
20) is equivalent to introducing the 4-lepton operator[c LL ] 1221 v σµ 2 ¯ 2 σµ 1 ,with the Wilson coefficient [c LL ] 1221 = .The constraint on this Wilson coefficient from the Z-pole observables can be read off using the global likelihood function quoted in Ref.[69].If only this one operator affects the Z-pole observables, the constraint reads −0.8 × 10 −3 < [c LL ] 1221 < 2 × 10 −3 at 95% CL.The resulting constraints on the parameter of our model are weaker than the ones from off-Z-pole measurements in LEP-2.

Table 1 :
The field content and U (2) F quantum numbers.
The consequence is that the muon and tau Yukawa couplings are set by the 23 and 32 off-diagonal elements.Indeed, diagonalizing Eq. (2.17) yields the Yukawa couplings y In summary, the charged lepton masses can be well reproduced with the U (1) F charge assignment in Eq. (2.16).The resulting structure of the Yukawa matrix in Eq. (2.17) leads to a large LH mixing between the 2nd and 3rd generation, s Le 23 ≈ 1, and a small RH 2-3 rotation, s Re 23 ≈ 0. The remaining freedom is the charge X E 1 and the three O(1) coefficients h e 12 , h e 13 , h e e ≈ h e 11 2 φ χ , y µ ≈ φ χ h e 32 , y τ ≈ h e 23 φ .(2.18) Using φ,χ in Eq. (2.14), the correct lepton masses are recovered by fixing three O(1) coefficients as h e 11 ≈ 0.57, h e 23 ≈ 0.29, h e 32 ≈ 4.3.The rotation angles are then determined by the remaining O(1) coefficients:

Table 4 :
Upper bounds on Wilson coefficients from leptonic and semi-leptonic K and B decays with s → d and b → d transitions.
+ τ − measured at the center-of-mass energies √ s ∈ [130, 207] GeV, as well as the differential cross-sections of e + e − → e + e − at √ s ∈ [189, 207] GeV.This way we obtain the 95% CL constraint: Ld 23 , of order 2-4 |V cb | .This leads to some tension with the bound from CP violation in kaon mixing in Eq. (4.4), assuming O(1) phases entering C K 1 .Much as the LFV bound, these constraints point to rather low k ≈ 2.