Minimal Lepton Flavor Violation Implications of the $b\to s$ Anomalies

The latest measurements of rare $b\to s$ decays in the LHCb experiment have led to results in tension with the predictions of the standard model (SM), including a tentative indication of the violation of lepton flavor universality. Assuming that this situation will persist because of new physics, we explore some of the potential consequences in the context of the SM extended with the seesaw mechanism involving right-handed neutrinos plus effective dimension-six lepton-quark operators under the framework of minimal flavor violation. We focus on a couple of such operators which can accommodate the LHCb anomalies and conform to the minimal flavor violation hypothesis in both their lepton and quark parts. We examine specifically the lepton-flavor-violating decays $B\to K^{(*)}\ell\ell'$, $B_s\to\phi\ell\ell'$, $B\to(\pi,\rho)\ell\ell'$, and $B_{d,s}\to\ell\ell'$, as well as $K_L\to e\mu$ and $K\to\pi e\mu$, induced by such operators. The estimated branching fractions of some of these decay modes with $\mu\tau$ in the final states are allowed by the pertinent experimental constraints to reach a few times $10^{-7}$ if other operators do not yield competitive effects. We also look at the implications for $B\to K^{(*)}\nu\nu$ and $K\to\pi\nu\nu$, finding that their rates can be a few times larger than their SM values. These results are testable in future experiments.


I. INTRODUCTION
The recently acquired data on a number of observables in b → s decays have revealed some intriguing tensions with the expectations of the standard model (SM). Specifically, last year the LHCb Collaboration [1] determined the ratio of branching fractions of the decays B + → K + µ + µ − and B + → K + e + e − to be R K = 0.745 +0.090 −0.074 (stat) ± 0.036(syst) for the dilepton invariant mass squared range of 1-6 GeV 2 . This result diverges from the lepton universality in the SM by 2.6σ. Moreover, earlier LHCb [2] reported a local discrepancy at the 3.7σ level from the SM prediction for one of the angular observables in the decay B 0 → K * 0 µ + µ − . This disagreement has persisted after an updated analysis was done using the full LHCb Run I dataset [3]. In addition, the latest measurements by LHCb [4] of the branching fractions of several rare b → s decays favor values less than those estimated in the SM.
Although the statistical significance of these anomalies is still too low for a definite conclusion, they may be hinting at the presence of physics beyond the SM. Subsequent model-independent theoretical works have in fact shown that new physics (NP) could resolve the tensions [5][6][7][8][9]. 1 In particular, NP contributing via dimension-six operators of the form [6][7][8] L SM+NP = α e G F V * ts V tb √ 2 π s γ β P L b ℓ γ β C ℓ 9 + C ℓ 10 γ 5 ℓ + H.c. (1) can produce one of the best fits to the b → s data if the Wilson coefficients C ℓ i = C SM i + C ℓ,NP i contain NP effects mainly in C µ,NP 9,10 which satisfy the condition C µ,NP 9 = −C µ,NP 10 ∼ −0.5. In this Lagrangian, α e and G F denote the usual fine structure and Fermi constants, V ts,tb are elements of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix, P L = (1 − γ 5 )/2, and at the m b scale C SM 9 ≃ −C SM 10 ≃ 4.2 universally for all charged leptons. In contrast, dimension-six leptonquark operators with tensor structures are excluded by the measurements and (pseudo)scalar operators can explain them only with much fine-tuning [6], while the counterpart of L SM+NP with right-handed quark chirality leads to much poorer fits to the data [7,8].
In view of the lepton nonuniversal nature of C ℓ,NP i and the size of C µ,NP i relative to C SM i , if the tentative indications of NP are substantiated by upcoming experiments, one generally expects that there can be b → s transitions which violate lepton-flavor symmetry and have rates within reach of searches in the near future [11]. 2 Such a possibility for lepton flavor violation (LFV), and other LFV phenomena that could have connections to the interactions of concern, have been examined further in the literature in the contexts of various NP scenarios [13,14].
In this paper, we also take these anomalies in b → s data to be due to NP and explore some of the potential consequences for a variety of rare meson decays with LFV. To do so, we adopt the framework of so-called minimal flavor violation (MFV), which is based on the hypothesis that Yukawa couplings are the only sources for the breaking of flavor and CP symmetries [15,16], as flavor-dependent quark interactions beyond the SM are empirically ruled out if they cause substantial flavor-changing neutral currents. Although the implementation of the MFV principle for quarks is straightforward, there is no unique way to extend it to the lepton sector, as the SM alone does not accommodate LFV and it is still unknown whether neutrinos are Dirac or Majorana particles. Since there is now compelling evidence for neutrino masses and mixing [17], it is interesting to formulate leptonic MFV by incorporating ingredients beyond the SM that can account for this observation [18]. Thus, here we consider the SM expanded with the addition of three heavy right-handed neutrinos as well as effective dimension-six quark-lepton operators with MFV built-in. The heavy neutrinos participate in the usual seesaw mechanism to endow light neutrinos with Majorana masses. We will focus on a couple of such operators which can bring about the NP contributions mentioned earlier and satisfy the MFV criterion in both the quark and lepton sectors. 3 We will examine how these operators may contribute to a number of rare b → s and b → d decays with LFV as well as the rare kaon decays K L → eµ, K → πeµ, and K → πνν.
In the next section, after briefly reviewing the MFV framework, we introduce the NP operators of interest. In Section III we write down the decay amplitudes and proceed with our numerical analysis. We provide our conclusions in Section IV. Additional information and lengthy formulas are relegated to an appendix.

II. OPERATORS WITH MINIMAL FLAVOR VIOLATON
In the SM supplemented with three right-handed Majorana neutrinos, the renormalizable Lagrangian for fermion masses can be written as where summation over k, l = 1, 2, 3 is implicit, Y u,d,ν,e are matrices for the Yukawa couplings, Q k,L (L k,L ) denote left-handed quark (lepton) doublets, U l,R and D l,R ν l,R and E l,R represent right-handed up-and down-type quarks (neutrinos and charged leptons), respectively, H stands for the Higgs doublet,H = iτ 2 H * with τ 2 being the second Pauli matrix, M ν is a matrix for the Majorana masses of ν l,R , and ν c k,R ≡ (ν k,R ) c , the superscript referring to charge conjugation. With the nonzero elements of M ν chosen to be much greater than those of vY ν / √ 2, the seesaw mechanism becomes operational [22], which leads to the light neutrinos' mass ma- GeV is the Higgs's vacuum expectation value, U PMNS denotes the Pontecorvo-Maki-Nakagawa-Sakata (PMNS [23]) matrix, and m ν = diag m 1 , m 2 , m 3 contains the light neutrinos' eigenmasses m 1,2,3 . This allows one to pick the form [24] where O is in general a complex matrix satisfying OO T = 1 1, the right-hand side being a 3×3 unit matrix.
We assume that the right-handed neutrinos are degenerate in mass, The MFV hypothesis [16,18] then implies that L m is formally invariant under the global flavor This entails that Q k,L , U k,R , D k,R , L k,L , ν k,R , and E k,R belong to the fundamental representations of their respective flavor groups, where V Q,U,D,L,E ∈ SU(3) Q,U,D,L,E and O ν ∈ O(3) ν is an orthogonal real matrix [16,18,19].
Furthermore, under G f the Yukawa couplings transform in the spurion sense according to Taking advantage of the symmetry under G f , we can work in the basis where and the fermion fields U k , D k ,ν k,L , ν k,R , and E k refer to the mass eigenstates. More explicitly, . We can then express Q k,L , L k,L , and Y u in relation to the CKM matrix V CKM and U PMNS as To put together effective Lagrangians beyond the SM with MFV built-in, one inserts products of the Yukawa matrices among the relevant fields to construct G f -invariant operators that are singlet under the SM gauge group [16,18]. Of interest here are the matrix products where y f = √ 2 m f /v. Since the biggest eigenvalues of A q and B q are, respectively, y 2 t ∼ 1 and y 2 b ∼ 3 × 10 −4 at the W -boson mass scale, for our purposes we can devise objects containing at most two powers of A q and neglect contributions with B q , as higher powers of A q can be related to lower ones by means of the Cayley-Hamilton identity [25]. As for A ℓ , to maximize the NP effects we assume that the right-handed neutrinos' mass M is sufficiently large to make the maximum eigenvalue of A ℓ equal to unity, which fulfills the perturbativity requirement [21,25]. Hence, as in the quark sector, we will keep terms up to order A 2 ℓ and drop those with B ℓ , whose elements are at most y 2 τ ∼ 1 × 10 −4 . Accordingly, the pertinent building blocks are where in our model-independent approach ζ 0,1,2 and ξ 0,1,2 are free parameters expected to be at most of O(1). Hence one or more of them may be suppressed or vanish, depending on the underlying theory. Since these parameters have negligible imaginary components [21,25], we can make the approximations ∆ † q = ∆ q and ∆ † ℓ = ∆ ℓ . It follows that the G f -invariant dimension-six operators which are SM gauge singlet and of the type that can readily give rise to the NP terms in Eq. (1) are where the objects ∆ qj and ∆ ℓj are, respectively, of the same form as ∆ q and ∆ ℓ in Eq. (11), but have their own independent coefficients ζ rj and ξ rj , and a = 1, 2, 3 is implicitly summed over. These generalize operators that were previously introduced under the assumption of MFV only in the quark [16] or lepton [18] part. The MFV effective Lagrangian of interest is then where the mass scale Λ characterizes the heavy NP underlying these interactions. In general, one could also consider dimension-six (pseudo)scalar or tensor operators, or operators with a righthanded quark current, that fulfill the MFV requirements and contribute to b → sℓl ′ transitions. However, since as discussed in Section I the current data do not favor such types of operators, their contributions can be neglected.
Before going into further details, it is instructive to go over the flavor structures of O 1 and O 2 . Expanding them in terms of the upper and lower components of the left-handed doublets Q k and L k and suppressing the gamma matrices, we arrive at with the y u,c terms having been dropped, summation over k, l, m, n = 1, 2, 3 is implicit, and ν k = (U PMNS ) knνn represents a flavor eigenstate. Hence, in the approximations we have made, L MFV does not cause flavor-changing transitions between up-type quarks. In contrast, flavor changes among downtype quarks can occur with either charged leptons or neutrinos being emitted, but according to Eq. (14) the two operators contribute differently to the two types of processes, which will be treated in more detail below. We will especially deal with the exclusive decays of b → qℓl ′ and b → qνν ′ for q = s, d and ℓ = ℓ ′ , as well as s → dℓl ′ and s → dνν ′ .

III. DECAY AMPLITUDES AND NUMERICAL ANALYSIS
In the presence of L MFV , one can generalize Eq. (1) to where λ qb = V * tq V tb is the CKM factor, q = s, d, and having set C SM 10 = −C SM 9 . Accordingly, for the terms with y u,c inξ rj having been dropped. It follows that From the foregoing, we can derive the contributions of L MFV to the amplitudes for a number where p B,K andṗ = p ℓ + p ℓ ′ denote the four-momenta of the mesons and dilepton, respectively, and F 0,1 stand for the form factors in the hadronic matrix element K sγ η (1 − γ 5 )b B , which are described in Appendix A and depend on the Lorentz-invariantṡ =ṗ 2 . ForB →K * ℓ − ℓ ′+ we arrive at where V and A 0,1,2 are the form factors for K * sγ η (1−γ 5 )b B , which are defined in Appendix A. We look atB s → φℓ − ℓ ′+ as well, which has an amplitude MB s→φℓl ′ analogous to where f Bs is the B s decay constant, which is also defined in Appendix A.
The MFV Lagrangian in Eq. (13) generates lepton-flavor-violating b → dℓ − ℓ ′+ processes with the same coefficient c ℓℓ ′ in Eq. (16), but a different CKM factor, λ db . This makes it of interest to include them in our study, particularlyB → πℓ − ℓ ′+ ,B → ρℓ − ℓ ′+ , andB d → ℓ − ℓ ′+ . Generally, there could be other dimension-six MFV operators that also contribute to b → dℓ − ℓ ′+ and therefore can enhance or reduce the impact of c ℓℓ ′ . Hereafter, we focus on the possibility that the effects of such operators are unimportant. Under these assumptions, the rates of these decay channels have expressions similar in form to those forB → Kℓ − ℓ ′+ ,B →K * ℓ − ℓ ′+ , andB s → ℓ − ℓ ′+ , respectively, but are comparatively smaller because |λ db /λ sb | 2 ∼ 1/22. Before starting our numerical calculation, we need to specify our choices further. Since the A ℓ matrix in Eq. (10) can be realized in many different ways, we concentrate on the least complicated possibility that O is a real orthogonal matrix, in which case For U PMNS , we employ the standard parametrization [17], with its elements being determined from the results of a recent fit to global neutrino data in Ref. [26], which depend on whether neutrino masses have a normal hierarchy (NH), m 1 < m 2 < m 3 , or an inverted one (IH), m 3 < m 1 < m 2 .
Since the empirical information on the absolute scale of m 1,2,3 is still far from precise [17], for definiteness we pick m 1 = 0 (m 3 = 0) in the NH (IH) case. Requiring the biggest eigenvalue of A ℓ to be unity then implies M ≃ 6.1 × 10 14 GeV. For the elements of V CKM , we adopt the results of the latest fit performed in Ref. [27]. With these numbers, we can evaluate the branching fractions of the decay modes discussed earlier using the rate formulas collected in Appendix A, which also describes our choices for the relevant hadronic form factors and decay constants.
We now attempt to attain the largest branching fractions of the b → qℓl ′ decays of interest, for ℓ = ℓ ′ , under our MFV framework by scanning the space of theξ + r parameters, which enter the NP term, c ℓℓ ′ , in the Wilson coefficient C ℓℓ ′ according to Eqs. (16)- (18). This amounts to maximizing |c ℓℓ ′ |. Simultaneously we need to impose the pertinent restrictions from the existing b-meson data. Thus, based on the results of an analysis of the latest b → s measurements including the LHCb anomalies [7,8], we require which can lead to one of the best fits to the data [8]. Nevertheless, it is possible to let c ee have some nonvanishing value as well [7]. Accordingly, we alternatively impose which is well within the allowed 1σ best-fit region in the second plot of Fig. 6 in Ref. [7].
In addition, since c ℓℓ ′ for a specific pair of ℓ and ℓ ′ = ℓ affects also s → dℓl ′ processes, we need to take into account the available experimental bounds, which may imply complementary limitations on c ℓℓ ′ . Indeed, if like before other dimension-six MFV operators do not give rise to competitive effects on s → dℓl ′ , we find in kaon data that the branching-fraction limit [17] B(K L → e ± µ ∓ ) exp = B(K L → e + µ − ) + B(K L → e − µ + ) exp < 4.7 × 10 −12 can translate into the strongest restriction on c ℓℓ ′ among lepton-flavor-violating meson decays. Thus, applying Eq. (A8) in the appendix, with the central values of the input parameters, we then extract which we will also impose. In our scans of theξ + r parameter space to maximize |c ℓℓ ′ |, we find that the bound in Eq. (26) is always reached. This implies that it can already be used to estimate the largest branching fractions of variousb →qeµ decays and K → πeµ within our MFV scenario with O 1,2 taken to be the main operators responsible. We display the results in the third column of Table I and compare them with the corresponding experimental limits if available. 4 One observes that the predicted B(B → Ke ± µ ∓ ) is only 4 times below its measured bound and, therefore, may be probed in near-future searches. For the other modes, the predictions are lower than their experimental counterparts by more than an order of magnitude.
It is evident from Table II that for each decay mode the results in the different cases are roughly of similar size and differ from each other within only factors of 4 or less. More interestingly, we notice that a few of the predicted branching fractions of the µτ channels can be as high as a few times 10 −7 . Although they are still at least about two orders of magnitude below the existing empirical limits, which are presently not many, upcoming experiments will expectedly offer ample opportunities to look for these decays and improve the data situation.
Since the restraint in Eq. (26) would lessen if there were other operators having destructive interference with c eµ in the K L decay amplitude, their presence would bring about a different set of predictions for the processes listed in Tables I and II. Changes in the predictions would also occur if other operators significantly modified b → dℓl ′ . Thus, our scenario in which O 1,2 dominate these LFV transitions will be tested when one or more of them are discovered and the acquired data compared to the predictions.
It is worth commenting, in addition, that the parameter values which are responsible for the predictions above also translate into reductions in the rates of the lepton-flavor-conserving b → (s, d)τ + τ − decays by up to a few tens percent with respect to their SM estimates. This implies that future observations of these processes with good precision will serve as important complementary tests on our NP scenario. Now, as indicated in the preceding section, L MFV in Eq. (13) also contributes to transitions with neutrinos in the final states. Specifically for b → qνν ′ decays the effective Lagrangian is given by where C SM L ≃ −6.4 is the SM prediction [13] and c νν ′ arises from NP. From L MFV , one then gets for νν ′ = ν k ν l c ν k ν l =c wherec (1,2) E k E l also enter c E k E l in Eq. (17), but with the opposite relative sign, andξ − r =ξ r1 −ξ r2 . Therefore, c νν ′ and c ℓℓ ′ are generally independent of each other [13].
Since only the contributions of c ν k ν l with k = l = 1, 2, 3 can interfere with the SM contribution to b → qνν, and since the neutrinos are not detected, it is straightforward to derive the ratio of branching fractions These channels are not yet observed, but there are experimental limits on their branching fractions [29]. Here the relevant bound is [13] B(B →Kνν) exp < 4.3 B(B →Kνν) SM , and so we can impose Their counterparts in the kaon sector, K + → π + νν and K L → π 0 νν, are similarly affected by c νν ′ . In its presence, the effective Lagrangian for sdνν ′ interactions is given by where λ t = λ sd , s 2 w = sin 2 θ W = 0.231, the SM term [30] X SM = X + |V us | 4 Re V * cs V cd P c /λ t consisting of 2 contributions from top-and charm-loop diagrams, respectively, with X = −C SM L s 2 w and |V us | 4 Re V * cs V cd = λ 7 /2 − λ 5 in the Wolfenstein parametrization, and x νν ′ = −c νν ′ s 2 w due to NP. The latest predictions of the SM are [31] B(K + → π + νν) SM = (9.11 ± 0.72) × 10 −11 and B(K L → π 0 νν) SM = (3.0 ± 0.3) × 10 −11 . Generalizing the expressions for these branching fractions to include the L MFV effects, which modify the X part, we obtain where κ ′ + = 0.997 × 5.173 × 10 −11 , X = 1.481, P c = 0.404, and κ L = 2.231 × 10 −10 are the central values from Ref. [31]. In view of the data [32] B(K + → π + νν) exp = 17.3 +11.5 −10.5 × 10 −11 and [33] B(K L → π 0 νν) exp < 2.6 × 10 −8 , we may then demand only In addition, comparing Eqs. (29) and (33), one infers that r B→K ( * ) νν is equal to its K L → π 0 νν counterpart, r K L →π 0 νν . For the parameter values that yield the examples in Table II, if c (2) ℓℓ ′ = 0 in Eqs. (17) and (28), we find that r B→K ( * ) νν = r K L →π 0 νν can be as large as 1.22 (1.15) in the NH (IH) case, whereas r K + →π + νν can reach 1.15 (1.11). If c (1) ℓℓ ′ = 0 instead, the results above for the decay modes with charged leptons are unchanged, but now the branching fractions of the modes with neutrinos tend to be reduced by up to 14% with respect to their SM expectations. All of these numbers are well within their corresponding restrictions in Eqs. (30) and (34).
If both c (1) ℓℓ ′ and c (2) ℓℓ ′ are nonzero, then in general c ℓℓ ′ and c νν ′ are not connected and, consequently, they can be maximized independently. In that case, our results in Table II are still the same, but for the channels with neutrinos, after scanning theξ − r parameter space subject to Eqs. (30) and (34), we obtain a maximum value of r B→K ( * ) νν = r K L →π 0 νν that saturates its limit of 4.3 and r K + →π + νν that reaches ∼ 3.1, withξ − 0 ∼ −7 and |ξ − 1,2 | ∼ 0. Thus future measurements can offer significant checks on these predictions.
Finally, we mention that, in the approximations we made, L MFV produces vanishing effects on the SM-dominated transitions b → (u, c)νℓ, s → uνℓ, and c → (d, s)νl. Although its contribution to t → bνl is nonzero, we estimate that its branching fraction is only under 10 −3 for optimistic choices of the parameter values, and so it is much smaller than B(t → bW + → bνℓ + ) SM ∼ 0.1 which is consistent with the data [17].

IV. CONCLUSIONS
We have entertained the possibility that the anomalies recently detected in the measurements of rare b → s processes are NP signals and explored some of the potential implications for a number of exclusive lepton-flavor-violating meson decays. Adopting the effective theory framework of MFV based on the SM plus 3 heavy right-handed neutrinos participating in the seesaw mechanism, we concentrate on a couple of dimension-six 4-fermion operators which accommodate the interactions that can yield one of the best fits to the b → s data. Assuming that these operators conform to the MFV hypothesis in both their quark and lepton parts, we evaluate their effects on various meson decays that violate lepton-flavor symmetry, subject to additional relevant b-meson and kaon constraints. For simplification, we further assume that other dimension-six MFV operators do not induce competitive contributions to these transitions. This scenario can be tested when they are discovered and the resulting predictions confronted with the acquired data.
With the preceding premises, our numerical work shows that for decays with charged leptons in the final states c eµ is the most restricted among the Wilson coefficients c ℓℓ ′ for ℓ = ℓ ′ , its strictest bound being supplied by the experimental limit for K L → eµ. Interestingly, among the other modes considered, the resulting prediction for B(B → Keµ) is the closest to its empirical bound, being only 4 times smaller, and therefore may be probed by forthcoming searches. Moreover, the predicted branching fractions of the exclusive b → sµτ decays can be as large as a few times 10 −7 and hence may be examined as well in near-future experiments. For decay channels with neutrinos in the final states, B → K ( * ) νν and K → πνν, we find that they are comparatively far less restricted and that the impact of the MFV operators can enhance their branching fractions by up to a few times with respect to the SM expectations. Thus, planned measurements on these processes involving neutrinos will provide important checks on the enhancements. according to [34] F 0 = a 00 + a 01 z + a 02 z 2 + a 03 z 3 , F 1 = a 10 + a 11 z + a 12 z 2 − 1 3 a 11 − 2a 12 z 3 1 −ŝ/ m B + 0.04578 GeV 2 , the a's being constants. Numerically, we adopt a 00 , a 01 , a 02 , a 03 = (0.54, −1.91, 1.83, −0.02) and a 10 , a 11 , a 12 = (0.43, −0.67, −1.12), which are their central values from Ref. [7]. For theB → π form-factors, which are relevant toB → πℓl ′ and defined analogously to F 0,1 in Eq. (A1), we make use of the parametrization choice preferred in Ref. [35] as well as the relation π − |dγ η b|B − = √ 2 π 0 |dγ η b|B 0 based on isospin symmetry. Similarly, forB →K * ℓl ′ the requisite matrix elements are wherep = p B − p K * and the form factors V and A 0,1,2 are functions ofs =p 2 . ForB s → φℓl ′ , the hadronic matrix elements have expressions similar to those forB →K * ℓl ′ . In numerical work, we adopt theB →K * andB s → φ form-factors available in Ref. [36] from combined fits to light-cone-sum-rule and lattice results. For the b → d transitionB → ρℓl ′ , the form factors are defined analogously to those in Eq. (A3), we again utilize the parametrization provided in Ref. [36], and as in theB → π case we have ρ − |dγ η (1 − γ 5 )b|B − = √ 2 ρ 0 |dγ η (1 − γ 5 )b|B 0 . The formulas in the last two paragraphs lead us to the amplitudes in Eqs. (19) and (20), respectively. Subsequently, we arrive at the differential rates whereṡ = (p ℓ + p ℓ ′ ) 2 , g = K m 2 B , m 2 K * ,ṡ , K(x, y, z) = x 2 + y 2 + z 2 − 2(xy + yz + xz) .