Clues for flavor from rare lepton and quark decays

Flavor symmetries successfully explain lepton and quark masses and mixings yet it is usually hard to distinguish different models that predict the same mixing angles. Further experimental input could be available, if the agents of flavor breaking are sufficiently low in mass and detectable or if new physics with non-trivial flavor charges is sufficiently low in mass and detectable. The recent hint for lepton-nonuniversality in the ratio of branching fractions $B \to K \mu \mu$ over $B \to K e e$, $R_K$, suggests the latter, at least for indirect detection via rare decays. We demonstrate the discriminating power of the rare decay data on flavor model building taking into account viable leptonic mixings and show how correlations with other observables exist in leptoquark models. We give expectations for branching ratios $B \to K \ell \ell^\prime, B_{(s)} \to \ell \ell^\prime$ and $\ell \to \ell^\prime \gamma$, and Higgs decays $h \to \ell \ell^\prime$.


I. INTRODUCTION
The LHCb collaboration [1] has presented recent data on lepton-nonuniversality (LNU): R K , the ratio of branching fractions B → Kµµ over B → Kee, R K has been measured as R LHCb K = 0.745 ± 0.090 0.074 ±0.036 (1) in the dilepton invariant mass squared bin 1 GeV 2 ≤ q 2 < 6 GeV 2 , which represents a deviation of 2.6σ from lepton universality [2]. In view of its generic leptoquark interpretation [3] we investigate directions in flavor model building that link the electron-muon non-universality to other flavor observables particularly lepton masses and mixing and lepton-flavor violation (LFV). In general if LNU is present, LFV can be expected, too [4]. R K has also been addressed within (composite) leptoquark models [5]; they do not aim at explaining LFV. Also recently, [6] considered scalar leptoquark contributions to rare leptonic B meson decays.
Here we consider frameworks where both phenomena (LNU, LFV) stem from the same (scalar) sector. To be more specific, we investigate the possibility of simultaneously accommodating LNU as observed through R K , and LFV, as observed in neutrino mixing and as hinted at 2.5 σ by CMS in h → τ µ [7]. We do so by adapting frameworks with non-Abelian flavor symmetries that predict the leptonic mixing matrix.
Due to the Standard Model (SM) gauge group, the maximal flavor symmetry is an U (3) for each chiral quark and lepton species. This observation motivates the use of SU (3) F or subgroups. After discussing briefly a framework of leptoquarks within the continuous group SU (3) F , [8] we consider explicit realizations with the discrete group A 4 in frameworks within supersymmetry (SUSY) [9][10][11][12].
While the specific results are related to the details of the models, our general conclusion is generic even if other discrete subgroups of SU ( The paper is organized as follows: in Section II we identify viable, data-driven patterns for flavor structure of the leptoquark to fermions coupling matrix. Predictions for rare LFV decays of leptons and B-mesons are given in Section III. The results of these sections are rather independent of flavor models. In Section IV these benchmarks are compared to what is realized in theoretical flavor models, where we consider models based on SU (3) F and A 4 known for explaining lepton flavor.
Rare Higgs decays are addressed in Section V. In Section VI we conclude. In several appendices we give formulae and auxiliary information.
In view of the uncertainties in the current data in the following we neglect factor of 1/2 differences in the semileptonic 4-fermion operators after Fierz-ordering and other small differences between the RL and LL model variants. This way the phenomenological constraints on the leptoquark parameters relevant to our study give identical hierarchies in flavor patterns, which we hence study jointly. We stress that despite the present day patterns being similar, there are differences in terms of model-building. The SU (2)-doublet leptoquark model couples to right-handed quarks 1 L = −λ q ∆ (qP L ) + h.c. , (2) while the SU (2)-triplet leptoquark model couples to left-handed quarks In each case, the couplings to the leptons are left-chiral and quark (q) and lepton ( ) denote flavor (generation) indices. We denote the mass of the leptoquark by M .
We continue discussing constraints on the generalized Yukawa λ q for q = d, s, b and = e, µ, τ : R K implies at 1 σ [3]: We restrict our analysis to M 1 TeV to be conservative with collider bounds [14][15][16][17], hence, in view of Eq. (5), |λ se λ * be − λ sµ λ * bµ | 2 · 10 −3 . There is also a contribution to B s mixing from a box diagram with leptons and leptoquarks in the loop giving rise to an operator of the formbsbs with the complex coefficient ( λ s λ * b ) 2 /(16π 2 M 2 ). Numerically, (λ se λ * be + λ sµ λ * bµ + λ sτ λ * bτ ) 2 M 2 (12 TeV) 2 , where the weaker bound is obtained from the mass difference |∆m NP s /∆m SM s | 0.15, whereas the stronger one in parentheses stems from the upper bound on the B s −B s mixing phase −0.015±0.035 (defined relative to the SM phase in the b → ccs decay amplitude) [18]. The complementarity of the |∆B| = 1 and |∆B| = 2 constraints allows to fix the mass scale and the dimensionless couplings separately, yielding an upper limit M O(50) TeV. Couplings to different quark flavors are within four orders of magnitude, 10 4 λ s /λ b 10 −4 [3].
The simplest scenarios are when the leptoquark couples to one generation of leptons only, as considered in [3]. In this case λ reads, A third variant with couplings to τ only cannot explain non-universality between electrons and muons as hinted currently by R K data. With either of those limits, λ [e] or λ [µ] , there is no LFV in rare processes. 2 Generalizing the single lepton scenario we consider the hierarchical pattern Here, we allow for quark flavor suppressions ρ d = λ d /λ b and ρ = λ s /λ b having larger couplings for heavier generations. This is suggested by the observed quark mass pattern; see Appendix B for details on how this is realized with the Froggatt-Nielsen (FN) mechanism [20] from U (1)-flavor symmetries. Here we simply take typical phenomenological profiles for Higgs-Yukawas of quark doublets (q) and down-type singlets (d) from [21], and apply them to the leptoquark-Yukawa, leading to where denotes a flavor parameter of the size of the sine of the Cabibbo angle, ∼ 0.2. We stress that what we mean by the hierarchies and flavor patterns in general is that they hold up to numbers (including CP phases) of order one. The overall scale λ 0 in our ansatz (λ [ρκ] ) depends on M and is fixed by R K through Eq. (5). 3 2 Generically RK < 1 can be explained by a suppressed branching ratio to muons, or an enhanced one to electrons, or combinations thereof, the details of which driven by an underlying model of flavor. Present data, however, is much more copious for rare decays into muons than into electrons due to the large data samples from the hadron machines at Fermilab and CERN LHC. This is likely to change in the nearer term future with the Belle II experiment taking data. As a consequence at present the constraints on the b → see modes are much more loose. In addition, hints for new physics in B → K * µµ angular distributions could point to a simultaneous explanation with RK , see [19] and references therein. So, the rather minimal scenario where RK originates at least predominantly from muon mode suppression and the electrons are SM-like presently has phenomenological support. 3 Note that here we do not explicitly specify U (1)F N charges for the leptons , whose assignments are not fixed by phenomenology as they depend on the model of neutrino masses. While in general the texture Eq.(8) with suppressions Eq.(9) is expected to hold, there is the possibility for cancellations between quark and lepton charges, that could lead to inverted hierarchies in the leptoquark Yukawa λ. We do not consider such solutions; they are absent as long as all charges are non-negative. observable current 90 % CL limit constraint future sensitivity B(µ → eγ) 5.7 · 10 −13 [22] |λ qe λ * qµ | M 2 (34TeV) 2 6 · 10 −14 [23] B(τ → eγ) a In previous versions of this work this bound was incorrectly stated as 1.2 · 10 −8 , which reflected the abstract of the original version of [24] in the arXiv, which has been corrected in June 2015.  [26] are for 50 ab −1 . For the constraint from B(τ → µη) we ignored the possibility of cancellations with λ dµ λ * dτ , see e.g., [30]. Following [31] we ignore tuning between leading order diagrams in the → γ amplitudes.
We emphasize that our main goal is to simultaneously explain lepton mixing and LNU. Lepton mixing is structurally very different from the hierarchical quark mixing matrix, which suggests a different origin of flavor for the leptons. We therefore consider in Section IV scenarios where the entries between columns of λ are related by factors that are ±1 or 0 as in Eq. (7). Such structures can occur naturally from a non-Abelian flavor symmetry distinguishing the generations of leptons.
We state the set of assumptions that lead to an effective factorization of quark flavor (rows) and lepton flavor (columns) in λ: the Higgs, leptoquark and quarks are singlets under the leptonic flavor symmetries while leptons are neutral under the quark symmetries. Most scenarios we consider will be factorized, including the hierarchical ansatz Eq. (8). Therein the flavor parameter κ allows to split the electrons from the other leptons, and is further discussed together with LFV below.
Although it is not completely general, the structure of λ [ρκ] is useful to obtain predictions, which we do in Section III, that can apply for more general flavor patterns, as shown in Section IV D.
Non-factorized scenarios are discussed in Section IV C 1 and Appendices D, E.
If couplings to more than one lepton flavor are present, existing constraints on LFV processes need to be considered. In Table I we collect LFV data, the corresponding constraints on λ-matrix elements, and give future sensitivities whenever available. Constraints on couplings to τ -leptons exist from the B-factory experiments Belle and Babar [24,25,27]. These bounds are not competitive with b-physics ones but can be improved in the near term future at Belle II [26]. The strongest constraint on λ qτ is therefore from B s -mixing Eq. (6). Consequently, phenomenology does not currently require a split (within our approximations) between muons and taus, which is why we refrain from adding a flavor factor separating the second and the third column of λ [ρκ] in Eq. (8).
Note also that while it would be straightforward to implement such a factor, for leptoquark masses near maximal, R K already requires muon couplings λ qµ of order one, cf. Eq. (5), so allowing λ qτ significantly larger than λ qµ would challenge perturbativity.
The strongest constraint on LFV stems from µ → eγ from MEG 2013 This is only mildly stronger than the R K bound Eq. (5), implying κ/ρ 0.5. The requirement of flavor hierarchies between electrons and muons can become more pronounced in the short-term future, with MEG upgrading their sensitivity to 6 · 10 −14 [23], which implies (κ/ρ) upgrade 0.2, if no signal is seen and R K remains close to its current determination. If the µ → eγ constraint tightens, it implies that in R K there has to be a dominant contribution to one lepton flavor with one branching ratio closer to its respective SM branching ratio than the other -ruling out solutions to Eq. (5) where e and µ couplings are of roughly the same order of magnitude.
Current limits on LFV in B-decays B(B → K ± ∓ ) are also given in Table I. They are weaker than Eq. (5). Comparison of the bound on B → Kµ ± e ∓ with the one on B → πµ ± e ∓ , also shown, implies ρ d /ρ 1.6.
Rare kaon decay data provide the strongest constraint on couplings to d-quarks [31]: Due to chiral symmetry contributions from purely left-chiral leptons to the anomalous magnetic moment of the electron or the muons ∆a ∼ |λ ql | 2 /(16π 2 )m 2 /M 2 are much smaller than current experimental sensitivities.
Note that the model by [5] with the leptoquark being an SU (2) L triplet has a flavor structure λ ij ∼ n(qi) n( j) , where the -powers are adjusted to give the correct fermion masses, similar to what could be obtained through a FN mechanism with U (1)-flavor symmetries. The resulting pattern turns out to be a viable subset of the one presented in Eq. (8). Specifically, ρ and ρ d follow Eq. (9), and κ ∼ m e /m µ ∼ 0.07. Couplings to muons are mildly suppressed relative to the ones to taus, of the order m µ /m τ ∼ 0.24, which is not far from our understanding of the symbol '∼' (within order one).
In Section IV we will encounter further viable patterns for λ which are not limiting cases of either single lepton flavor Eq. (7) or hierarchy Eq. (8).

III. LFV PREDICTIONS IN TERMS OF R K
The leptoquark framework with the λ [ρκ] ansatz from Eq. (8) is very predictive and allows for correlations between rare b-decays and LFV. We obtain and For the amplitudes of the purely leptonic decays note the m τ /m µ chiral enhancement of A(B s → τ + (µ, e) − ) over A(B s → τ − (µ, e) + ) and A(B s → µ + µ − ). More general, neglecting phase space, This relation follows from the left-handed lepton only hypothesis, a feature of the beyond SM (BSM) models considered here, see Appendix A for details. Eq. (20) can be used to test the lepton chirality in LFV processes.
In addition, for = , Precise predictions for lepton flavor conserving decays we leave for future work. We stress that the calculation of the LFV hadron decays is much less complicated as contributions of quark loops or (qq)-resonances coupling to the electromagnetic current are absent.
There are two extreme scenarios assuming λ 0 ∼ 1: , corresponding to high mass leptoquark in the few×10 TeV range, i.e., out of LHC reach. In this case κ can be of order one, too, implying that B → Kµ ± e ∓ and µ → eγ could be just around the corner, cf. Table I. Radiative τ -decays are far away while B(B → K(e, µ) ± τ ∓ ) and B(τ → µη) are at least three orders of magnitude away from their respective current limits.
B) ρ 1, corresponding to light leptoquarks up to few TeV in mass and κ 1. As in scenario In case of at least two more measurements rather than upper bounds on LFV in addition to R K the parameters ρ and κ can be determined, pinning down the flavor pattern of the leptoquark coupling matrix λ further. It is conceivable that the latter is linked to the mechanism generating flavor for SM fermions, e.g., with flavor symmetries, which can be probed with the rare decay data. In Section IV we give examples of realistic flavor symmetries that give the single lepton flavor patterns Eq. (7), and special cases of the hierarchy pattern Eq. (8), but also further testable pattern. Predictions of leptoquark coupling patterns not covered by the ansatz Eq. (8) are given in Section IV D.

IV. FLAVOR SYMMETRIES
In this section we illustrate how flavor symmetries generically control the shape of the leptoquark Yukawa coupling matrix λ, and how there is in general a relation to Higgs Yukawa couplings controlled by the same symmetry. The reason for this is simply that both the leptoquark ∆ and Higgs doublet (or doublets) couple to same fermions, whose generations transform as specific representations of the flavor symmetry.
We use specific models to demonstrate this, and show how to obtain some special cases for λ, from Eq. (7), and λ [ρκ] from Eq. (8), but also new structrures.
In terms of the quark index in λ, the structure between rows depends on what kind of symmetry transformations we assign to either the Q i generations (for the LL leptoquark) or to the RH downtype quarks d c i (for the RL leptoquark). In frameworks where the flavor symmetry distinguishes Q and d c , the LL and RL leptoquark models could be very different. Here, we focus mostly on frameworks of flavor symmetries explaining leptonic mixing, where this is not the case. If we assume all quarks are trivial singlets of whatever non-Abelian symmetry is included, one can still assign a FN charge to quarks (under U (1) F N ) leading to Eq. (9) or similar. The FN mechanism relies on having lighter generations with larger charges in order to explain the mass hierarchies through added insertions of a θ field. We choose without loss of generality to normalize the U (1) F N charge of θ to −1. If we assume that the leptoquark ∆ is neutral under the U (1) F N , the leptoquark Yukawa couplings to lighter generations of quarks is simply suppressed by as many additional insertions of θ as the ones that appear for the respective masses, resulting in Eq. (9). We relegate further considerations of the U (1) F N charges of ∆ to Appendix B, in order to focus on the structure of the lepton couplings, where we consider non-Abelian flavor symmetries.
Before we delve exclusively into lepton flavor models, it is interesting to consider a SUSY framework with an underlying SO(10) unified gauge group, where quark and leptons are linked. This is particularly restrictive in terms of the allowed flavor structures. At this level we will consider the flavor symmetry to be continuous (SU (3) F , Section IV A).
We then investigate in some detail a SUSY framework with discrete flavor symmetry A 4 in Sections IV B and IV C. We focus on A 4 because it is a convenient framework to obtain the observed pattern of leptonic mixing, but also due to its relative simplicity as the smallest group with triplet representations. Appendix C contains a brief primer on A 4 .
The reason for having SUSY in all the frameworks discussed in this section is twofold: first, SUSY keeps the gauge hierarchy problem under control, which is particularly relevant when going beyond the SM (with SO(10) or flavor symmetries broken at a high scale). Second, by holomorphy it allows one to separately align the different flavor symmetry breaking directions required in the respective models. Because of SUSY, we necessarily use two SU (2) doublets, h u and h d .
All the frameworks also have a U (1) R R-symmetry, which is spontaneously broken to its Z 2R subgroup acting like the MSSM R-parity. Another point that all the frameworks we consider have in common is the presence of an auxiliary Abelian flavor symmetry (either U (1) F , Z 3 or Z 4 ). As a point of notation, in each section the charge of a superfield φ under the relevant auxiliary symmetry is denoted as {φ}.
We discuss now a framework exemplified by the model presented in [8]. Similar models were considered in [37,38], showing how to obtain large reactor angle (θ 13 ). One of the main features in the framework are that the 3 generations of each fermion flavor transform as triplets of SU (3). 5 In order to build flavor symmetry invariants, we add three flavons (each with 3 generations) with b ∼ 0.20a, c ∼ 0.20b. This VEV hierarchy is required to be of order of the Cabibbo angle, but is also related to the hierarchy between the solar and atmospheric neutrinos [8] and even to the magnitude of the reactor angle θ 13 [38].
After SO (10)  The Yukawa couplings appear from non-renormalizable terms which are generally of the type h u,d and are controlled also by an auxiliary Abelian symmetry U (1) F . The VEV directions in Eq. (25) are responsible for giving hierarchical structures for all the charged fermions. 6 We focus now on the leptoquark Yukawa structures λ that occur naturally when leptoquarks are added to this framework. In order to study this, we need to keep in mind the transformation Note also that, due to the underlying unification d c and Q both transform equally under the flavor symmetry so there is no difference in the λ structures corresponding to a LL or RL leptoquark, and also that structures involving components from ψ (Q, L) and ψ c (d c ) are symmetric due to their origin from the same SO(10) multiplet. If we were considering a unified gauge group that is not left-right symmetric then the Q and d c would generally transform differently under the flavor symmetry and the structures for the RL and LL leptoquark would be different. A prime example of this is SU (5), where lepton doublets L belong to the same GUT multiplet as d c , but Q is in a different GUT multiplet together with u c and e c . See Appendix E for some possibilities.
which are very similar to those shaping the fermion masses. Which particular flavons couple to ∆ is determined by its U (1) F charge {∆}: which is a special limit of Eq. (8) where ρ = 1 and κ = ρ d = 0, given in Eq. (26).
As we intend to account for R K , we show only the structures with non-zero entries simultaneously in λ se , λ be or simultaneously in λ sµ , λ bµ .
The λ flavor structure for {∆} = −2 reads which is a highly predictive limit of λ [ρκ] Eq. (8) with the same exact magnitude on the 4 non-zero entries. As there is a single coupling we can absorb the order one coefficient by redefining λ 0 .
which due to a b c has ∆ couple predominantly to bτ . We kept x explicit to separate leading and non-leading contributions and redefine λ 0 to absorb the order one coupling of the subleading contribution, which, although suppressed by ∼ bc/(xa 2 ) could in principle account for R K . However, the simultaneous presence of the kaon constraint Eq. (11) rules this out.
The democratic pattern from {∆} = −6 is and very symmetric like λ [−2] , but it preserves lepton universality so it can not account for R K = 1.
The only viable texture is therefore λ is invariant, with the same texture as λ [−2] but with overall U (1) F charge +6 added due to and so on, all except Eq. (29) automatically vanish due to the Levi-Civita tensor.
The first A 4 framework we consider is of SUSY A 4 × Z 3 × U (1) F N models, initially proposed in [9] and with renormalizable UV completions [11] and in particular [12] which obtains non-zero reactor angle θ 13 in full agreement with current neutrino oscillation data. The FN mechanism [20] is implemented separately through U (1) F N , generating the hierarchy in the charged lepton masses without requiring small Yukawa couplings; it can easily be used to justify the quark hierarchies as it acts as R-parity. We refer to Table II for details about the charge assignments.
We have not listed the A 4 representation of ∆ in Table II as it requires a more detailed discussion.
As we have done for SU (3) F , we focus on the case where the leptoquark is an A 4 singlet. We consider A 4 triplet leptoquarks in Appendix D.
For completeness, we include here very briefly the charged lepton Yukawa couplings in the See text 1  1  1 3  3  1 1 1  1  1  1 3 3 1 1 3   superpotential where Λ is a scale associated with the breaking of the flavor symmetry. Coupling to the (1, 0, 0) VEV results in a diagonal mass matrix for the charged leptons where we can identify L 1 with e, L 2 with µ and L 3 with τ .
For ∆, the hypercharge differs for the LL and RL leptoquark. The Yukawa coupling to leptoquarks is associated with a renormalizable superpotential term that is either λ ij [d c i ∆L j ] for SU (2) doublet ∆ corresponding to Eq. (2) -or, with a SU (2) triplet ∆, one would have λ ij [Q i ∆L j ] as a SUSY version of Eq. (3). The quark generation index is i = d, s, b and the lepton generation index is j = e, µ, τ . We assume that all three generations of SU (2) L doublet and SU (2) L singlet quarks are trivial singlets of A 4 . We consider SU (2) L singlet down-type quarks as an A 4 triplet in Appendix E as this case arises in A 4 unified models [39,40] with SU (5).
The columns of λ are constrained because L is an A 4 triplet. If the leptoquark is an A 4 singlet, the renormalizable leptoquark superpotential terms are no longer A 4 invariant. A contribution to λ appears at leading order (LO) from a non-renormalizable term where L contracts with an A 4 triplet flavon. There are then three options for ∆ to transform under A 4 , and 4 non-equivalent ways to build an A 4 invariant: 4. φ i ν L i ∆ couples equally to all lepton generations due to φ ν .
By coupling to the VEV (1, 0, 0), the different options for the leptoquark representation under A 4 lead to a single non-vanishing column for λ. Options 1. and 2. correspond respectively to explicit realization of the patterns λ [e,µ] in Eq. (7), whereas option 3. gives SM-like R K and is disfavored by the current LHCb measurement. Option 4., which occurs regardless of ∆ ∼ 1, 1 , 1 , preserves lepton universality at LO but as discussed later has a myriad of LNU couplings already at NLO.
For now we postpone option 4. and take the Z 3 charge of ∆ to be {∆} = 2 (following notation used in other sections for Abelian charge). This choice means that at LO ∆ couples to SM fermions only through φ l .
This is a non-trivial result that we emphasize: using the same non-Abelian flavor symmetry A 4 and VEVs that jointly predict viable lepton mixing with large angles, one can obtain automatically leptoquark flavor structures where LNU exists due to the isolation of a single lepton generation. This is consistent with the LNU in B to K decays, as shown in [3], where the isolation of the e or This can be identified more precisely in specific UV completions [11,12], where it may turn out the isolation is actually exact (due to e.g. missing messengers for NLO diagrams). Generically one may still associate next-to-leading order (NLO) effects to the presence of non-zero reactor angle are phenomenologically viable. We conclude that LNU at NLO gives a viable pattern for λ.
One could consider assigning quarks non-trivially under A 4 , however, in the present framework this leads to issues in obtaining viable quark masses and mixing. Instead we will explore this option further in a different framework, A 4 × Z 4 , in Section IV C 1.
The A 4 × Z 4 SUSY framework [10] is another interesting A 4 framework. For its renormalizable UV completions see also [11], and [12] for considerations regarding large θ 13 .
We start with a brief comparison with the A 4 × Z 3 framework discussed in the previous section: the first difference is that the VEV of the flavon triplet coupling to the charged leptons is now φ l ∝ (0, 1, 0). The second one is with respect with the FN mechanism and fields -neither U (1) F N nor θ are present, with the charged lepton hierarchy being due to a field θ . 7 The last difference is that the sector neutral under Z 4 is the neutrino sector whereas with Z 3 , the charged lepton sector was neutral. This is particularly relevant due to e.g. the ξ flavon neutral under Z 4 . 8  When adding a leptoquark to this framework one can again obtain LO λ structures like Eq. (7) by having the Z 4 charge of ∆ be {∆} = 2. At LO we have: They are similar to options 1.,2.,3. of Section IV B, modified slightly to account for the (0,1,0) direction of the VEV.
More significant differences arise beyond LO, because the neutrino sector is neutral. Given that {φ ν } = 0, one is allowed to add it to the LO contribution thus constructing NLO contributions involving [φ l φ ν ] 3a,3s . This is similar to what we have seen in Section IV B, although the effective VEVs of the combinations are now φ 2 . In order to keep the model predictive, we assume now that the φ ν contributions to λ are forbidden by a partial UV completion where beyond LO terms are only allowed through singlet insertions (this is a natural consequence if A 4 triplet messengers are absent [12]). In this scenario the only relevant flavon beyond LO is ξ , which is also neutral under the Z 4 . NLO contributions come from one insertion of ξ and NNLO contributions from two insertions of ξ . The options for ∆ are: 3. ∆ ∼ 1 has LO coupling to µ, φ i l L i ξ ∆ NLO coupling to τ , φ i l L i ξ ξ ∆ NNLO coupling to electron.
We illustrate these in matrix form, defining κ ≡ ξ /Λ ∼ 0.2: where we also parametrized the quark suppression factors as in Eq. (8) for a more direct comparison.
The first pattern, λ [1] cannot simultaneously accommodate R K and B s -mixing constraints unless  What happens with different generations of quarks assigned as different A 4 singlets is that, depending on the quark generation, the leptoquark couples to different columns at each order. As an illustration of this, and neglecting the quark FN charges for simplicity, take ∆ ∼ 1 , d c 2,3 still as trivial singlets but d c 1 ∼ 1 . We have at LO: corresponding to: Such patterns generalize lepton flavor isolation Eq. (7) In general, constructions of this type yield patterns for λ LO where a single quark flavor can couple to a single lepton flavor. Due to R K , either electrons or muons have to couple to both s and b. Therefore, the following structures in addition to Eq. (34) are phenomenologically viable: All of these effective two-flavor patterns predict LFV involving two lepton flavors only at LO, but receive respective NLO (and NNLO) contributions which are similarly shifted in the first row and are also themselves a two-flavor pattern.
However, in specific UV completions it is possible to forbid NLO and NNLO contributions to λ, while allowing the desired LO contributions and the necessary Y d ij Yukawa couplings. 9 In fact, allowing only a minimal set of messengers we can obtain λ matrices like Eq. (34) with x 1 = 0, corresponding to the truly minimal patterns from [3] that explain LNU data Eq. (1).

D. Summary of flavor symmetry frameworks and predictions
We summarize in Table IV   The other genuine non-Abelian type of λ structure is the inverted hierarchy Eq. (32), λ [1 ] , in which the leptoquarks couplings do not follow the ones of the leptons to the Higgs. This pattern predicts LFV in processes involving eµ, eτ and µτ , which can be read off from the estimates given in Section III using κ = κ , κ = κ 2 and κ = κ 3 , respectively. The normal hierarchy pattern, λ [1 ] of Eq. (32), predicts LFV in processes involving eµ, eτ and µτ corresponding to κ = κ 2 , κ = κ 3 and κ = κ , respectively. In these cases, κ = O(θ 13 ).
While it appears that it is possible to obtain in each flavor model pattern for RL and LL leptoquarks alike, we stress that this is a feature of the specific frameworks considered where Q and d c transform equally under the flavor symmetry, either due to underlying unification or for the sake of simplicity.

V. RARE HIGGS DECAYS
In this section we discuss leptoquark effects in decays of the Higgs into fermion anti-fermion.
Such decays have received recent interest with the advent of such a particle and its various decay modes being stringent test of the flavor sector of the SM and beyond, e.g., recently, [42,43].
Leptoquark contributions to Higgs decays are induced at one-loop as exemplified in Figure 1.
The amplitude is proportional to a renormalizable term ∆ † ∆h † h, whose coefficient in general is model-dependent. After electroweak symmetry breaking a coupling of the Higgs to two leptoquarks is induced at order vh∆∆ * , where v 174 GeV sets the electroweak scale. The corrections to flavor-diagonal modes h → are hence parametrically given as where N c = 3 denotes the number of colors. Analogously for h → qq: Contributions from further one-loop diagrams involving the Higgs-Yukawa couplings exist but are suppressed further by at least y b,τ 1, and contributions with y t don't bring in additional enhancements either. The relative corrections δy/y to the diagonal Higgs Yukawas therefore do not exceed the 10 −2 level. Spurion analysis shows when LFV in the Higgs coupling¯ L Xe R is induced, cf. Eq. (2), hence the leptoquark couplings, in general, induce LFV couplings. A single Higgs doublet as in the SM suffices. Note that single lepton-flavor patterns as in Eq. (7) fail to induce X ij for i = j.
Leptoquark contributions to the B-term in the spurion expansion arise at order y is suppressed relative to y by y /y for m < m .
The coupling in Eq. (39) is the same which drives → γ. We find that, using the limits from Table I, LFV Higgs effects are limited as y µτ y τ 10 −3 , y eτ y τ 10 −3 , y eµ y µ 10 −6 .
Flavor models of course predict patterns for these couplings and relate h → decays. E.g., in terms of the parametrization Eq. (8): If κ is large, this implies that all LFV branching ratios are strongly suppressed due to the suppression of h → µe. If κ is small, the h → τ µ branching ratio is much larger than the ones involving electrons.
Different patterns follow from other flavor structures of λ summarized in Section IV D.

VI. CONCLUSIONS
The bottom-up BSM scenario with leptoquarks considered here provides further evidence of how new physics can help to learn about flavor [46]: if the anomalous signatures in the FCNCs Eq. (1) hold they offer unique possibilities to probe the mechanism of flavor, such as the type and charge assignment of a flavor symmetry whose imprints on SM matter were the only experimental information previously available.
By adding a leptoquark to existing flavor symmetry frameworks we obtained sample scenarios where the flavor symmetries are simultaneously responsible for LNU (R K ) and LFV, drawing in each case relations to the lepton mixing angles and charged fermion hierarchies predicted in these models.
The predictions for LFV lepton and b-decays are copious, and we summarize them in Sections III and IV D. The LFV branching ratios, whose size is driven by ( 7), is observed to below percent level, that points to models such as our A 4 × Z 3 which only allow deviations from isolation at the NNNLO -or, given an appropriate UV completion, to models like our A 4 × Z 4 . Mimicking these patterns without using non-Abelian symmetries would require rather unnatural leptoquark couplings.
If LNU and LFV in rare b-decays is observed, it allows to completely identify the chiralitystructure of the leptoquark-lepton-quark couplings: comparison of R K with related non-universality tests into others strange final states, such as K * , X s , ... allows to probe for right-handed quark currents [47]. Specifically, the models Eq. The current hint for BSM in |∆B| = 1 transitions in R K provides an anchor for fixing BSM scales when interpreted as a "signal" and combined with |∆B| = 2 (B s -mixing) bounds. The mass range of the leptoquarks is determined to be right above search limits around O(TeV) and below O(50)TeV [3]. In pp collisions the leptoquarks will be pair-produced and decay in our frameworks predominantly into second or third generation leptons and into third generation quarks.
Note added: During the completion of this work a paper [48], appeared, where related leptoquark effects to h → were considered.
where α e , V ij and G F denote the fine structure constant, the CKM matrix elements and Fermi's constant, respectively, the semileptonic operators receive leptoquark contributions as [3] Note that C where we used that the hadronic matrix element is proportional to the B s mesons' four-momentum, which equals the sum of the leptons' four-momenta q µ , and then applied the equations of motion for particle u and anti-particle spinors v. The model-independent framework Eq. (A1), Eq. (A4) allows to compute leptoquark effects in B → K ( * ) µµ decays in a straightforward way. Correlations of the global fits and R K in such models have been discussed recently in [3,5]. A detailed exploration including the very recent, preliminary 3fb −1 data by LHCb on B → K * µµ angular observables [49] taking into account SM uncertainties is beyond the scope of our work.

Appendix B: Quarks and Froggatt-Nielsen
Relative suppression between rows of λ is naturally obtained by assigning different U (1) F N charges to the generations of quarks. The outcome depends also on the charge of the leptoquark.
In this appendix curly brackets denote the U (1) F N charge of a field.
The leptoquark couplings would be: where = θ /Λ ∼ 0.2. x d,s,b are order one dimensionless couplings. The outcome is an hierarchy between rows of the leptoquark couplings λ d < λ s < λ b as e.g. λ d /λ s = x d xs , irrespective of lepton flavor. One can easily generalize this for alternative assignments of FN charges, such as  For triplet ∆ i , the invariants can be quite different from those discussed in section IV A. If {∆} = 0 there is an invariant not involving the flavons ijk ∆ i Q j L k . This is a purely anti-symmetric structure for each λ i : For this charge assignment there are no holomorphic leptoquark bi-linears to study, but we can nevertheless conclude that this type of structure can not account for R K regardless of the mass eigenstates.
If {∆} = 0, invariants can arise by contracting to 3 flavons, such as (φ i . It could give rise to a mass contribution after the R-symmetry is broken: Diagonalizing this mass matrix reveals ∆ 1 has vanishing mass prior to soft terms, so we expect it to remain as the lightest leptoquark -however it also has no coupling to SM fermions. The next lightest mass eigenstate with mass mass (−1 + √ 2)m ∆ ab, and the heaviest with mass (1 + √ 2)m ∆ ab, are respectively dominantly ∆ 2 (with some ∆ 3 ) and the orthogonal combination. As both ∆ 2,3 couple to SM fermions with the λ [−2] texture, this option reduces to a more involved version of the singlet leptoquark discussed before.
For anti-triplet ∆ i , the invariants require only a single flavon contraction (contrasting to fermion mass structures that required 2, and the anti-triplet which as we have seen requires either 0 or 3).
An example would be for {∆} = −1, where we have (∆ i Q i )(φ j 23 L j ) + (φ i 23 Q i )(∆ j L j ): There is no contribution from holomorphic bi-linears, as ijk φ i 23 ∆ j ∆ k vanishes. We can still conclude that these structures would allow for R K = 1 as long there are sufficiently light eigenstates containing ∆ 1,2 . Each generation has its own λ i matrix. The LO-structures are: (D5) The holomorphic bi-linears in ∆ allowed by where we absorbed the magnitudes of the respective VEVs into the coefficients x. This is similar to the Majorana neutrino structure in this framework (see [12]). With eigenvalue x ξ + x ξ , (1, 1, 1) is an eigenstate of this structure which, when compared to the respective λ i , would preserve lepton universality. In the limit ξ = 0 we have effectively x ξ = 0 and the other two eigenstates are also independent of the free parameters: they would be (2, −1, −1) and (0, 1, −1), respectively with eigenstates x ν + x ξ and x ν − x ξ . These two leptoquark eigenstates could mediate LNU couplings.
As it is ξ = 0 that generates non-vanishing reactor angle in this A 4 × Z 3 framework, in realistic regions of parameter space the (2, −1, −1) and (0, 1, −1) directions are no longer eigenstates of the leptoquarks. R K = 1 is still possible and an interesting situation arises where the perturbations away from the (2, −1, −1) and (0, 1, −1) directions to the leptoquark mass eigenstates are directly related to θ 13 = 0 and the required perturbations of the neutrino eigenstates, appearing in the leptonic mixing matrix (recall in this basis the charged leptons are diagonal).
Studying NLO corrections for the A 4 triplet ∆ i , which requires correcting both the λ i and the assumed structures for M ∆ , is beyond the scope of the present work.
Appendix E: d c as A 4 triplet We consider here a situation where Q remains as singlets under the flavor symmetry but d c is, like L, an A 4 triplet. This situation arises naturally in A 4 unified models of lepton mixing [39,40] with SU (