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 → Kμμ over B → Kee, RK, 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 → Kℓℓ′, B(s) → ℓℓ′ and ℓ → ℓ′γ, and Higgs decays h → ℓℓ′.


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 (1.1) 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 JHEP06(2015)072 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(3) F are invoked. If the same flavor symmetry breaking fields, often referred to as flavons, are involved in Higgs and leptoquark couplings, the respective Yukawa structures are related and there are links between what is predicted for LNU and LFV (in lepton mixing and Higgs decays).
The paper is organized as follows: in section 2 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 3. The results of these sections are rather independent of flavor models. In section 4 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 5. In section 6 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 JHEP06(2015)072 leptoquark model couples to right-handed quarks 1 L = −λ q ∆ (qP L ) + h.c. , (2.1) while the SU(2)-triplet leptoquark model couples to left-handed quarks L = −λ q ∆ * (q ) + h.c. . (2.2) 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 : We restrict our analysis to M 1 TeV to be conservative with collider bounds [14][15][16][17], hence, in view of eq. (2.4), |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 , M 2 (17.3 TeV) 2 , (2.5) 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, The RL model has similarities with the R-parity violating minimal supersymmetric standard model (MSSM); the corresponding term is the λ LQd c coupling. Note also that the ∆(3, 2) 1/6 with mass much lighter than the GUT-scale does not require further model-building to avoid proton decay [13].

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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. (2.4). 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 4 scenarios where the entries between columns of λ are related by factors that are ±1 or 0 as in eq. (2.6). 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. (2.7). 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 3, that can apply for more general flavor patterns, as shown in section 4.4. Non-factorized scenarios are discussed in section 4.3.1 and appendices D, E.
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.
If couplings to more than one lepton flavor are present, existing constraints on LFV processes need to be considered. In table 1 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. (2.5). 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. (2.7). 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. (2.4), 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. (2.4), 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. (2.4) 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 1. They are weaker than eq. (2.4). Comparison of the bound on B → Kµ ± e ∓ with the one on B → πµ ± e ∓ , also shown, implies ρ d /ρ 1.6.

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Rare kaon decay data provide the strongest constraint on couplings to d-quarks [31]: (2.10) 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.
To summarize, the phenomenologically viable range for λ [ρκ] parameters in eq. (2.7) is In view of µ → eγ, if ρ 1 (corresponding to a low mass leptoquark), then necessarily κ 1. Regarding µ → e conversion in nuclei, the current best limit Γ(µAu → eAu)/Γ(µAu) < 7 · 10 −13 [32] is consistent with the bounds (2.11), however, future experiments such as COMET [33] and Mu2e [34] with sensitivity below 10 −16 are sensitive to the parameter space of leptoquark models discussed here. 3 We note that corresponding tree-level contributions can always be avoided by decoupling down-quarks, suppressing ρ d , or decoupling electrons, suppressing κ. 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. (2.7). Specifically, ρ and ρ d follow eq. (2.8), 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 4 we will encounter further viable patterns for λ which are not limiting cases of either single lepton flavor eq. (2.6) or hierarchy eq. (2.7).

LFV predictions in terms of R K
The leptoquark framework with the λ [ρκ] ansatz from eq. (2.7) is very predictive and allows for correlations between rare b-decays and LFV. We obtain

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and B(µ → eγ) 2 · 10 −12 κ 2 ρ 2 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. (3.8) can be used to test the lepton chirality in LFV processes. Furthermore we obtain where B(B s → µ + µ − ) SM = (3.65 ± 0.23) · 10 −9 [35]. The current bound B(B s → µ ± e ∓ ) < 1.1 · 10 −8 at 90 % CL by LHCb [36] is hence at least two to three orders of magnitude away. 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.
1, corresponding to light leptoquarks up to few TeV in mass. This scenario arises for instance in the model in appendix B where the leptoquark carries FN charge to increase powers of spurion insertions. The LFV pattern follows from the values of ρ, ρ d , κ as dictated by the flavor model.
Concerning collider searches within pattern eq. (2.7) the leptoquark decays take place into second or third generation leptons and into third generation quarks. In scenario A), all combinations of quarks and leptons can arise at similar level except for those involving d-quarks, which are necessarily suppressed.
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 4 we give examples of realistic flavor symmetries that give the single lepton flavor patterns eq. (2.6), and special cases of the hierarchy pattern eq. (2.7), but also further testable pattern. Predictions of leptoquark coupling patterns not covered by the ansatz eq. (2.7) are given in section 4.4.

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.
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 down-type 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) FN ) leading to eq. (2.8) 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) FN charge of θ to −1. If we assume that the leptoquark ∆ is neutral under the U(1) FN , the leptoquark Yukawa couplings to JHEP06(2015)072 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. (2.8). We relegate further considerations of the U(1) FN 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 4.1).
We then investigate in some detail a SUSY framework with discrete flavor symmetry A 4 in sections 4.2 and 4.3. 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). 4 In order to build flavor symmetry invariants, we add three flavons (each with 3 generations) named φ i 3 , φ i 23 and φ i 123 (the subscript numbers are labels, the upper indices are generation indices i = 1, 2, 3, anti-triplet indices of SU (3)). Through the details of the superpotential of the model, the flavons acquire vacuum expectation values (VEVs), breaking the flavor symmetry in specific directions: 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) is broken to a left-right symmetric GUT containing SU(2) L × SU(2) R (before breaking to the SM), the left-handed fermions are referred as ψ i and the conjugates JHEP06(2015)072 of the right-handed fermions as ψ c j (where i, j = 1, 2, 3 are generation indices, triplet indices of SU(3)), where ψ contains the 3 generations of Q, L and ψ c contains the 3 generations of u c , d c , e c and ν c . The superfields containing the electroweak doublets we denote here as h u , h d .
The Yukawa couplings appear from non-renormalizable terms which are generally of the type (φ i 3 ψ i )(φ j 3 ψ c j )h u,d and are controlled also by an auxiliary Abelian symmetry U(1) F . The VEV directions in eq. (4.1) are responsible for giving hierarchical structures for all the charged fermions. 5 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 properties of the SM fermions, which are all triplets under SU(3) F , and the charges of the flavons under U(1) F , which we denote in terms of curly brackets e.g. 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.
If ∆ is an SU(3) F singlet (as h u,d ), the flavor symmetry invariants shaping λ are ∆(φ i Q i )(φ j L j ), 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. (2.7) where ρ = 1 and κ = ρ d = 0, given in eq. (4.2).

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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. (2.7) 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 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. (2.10) 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 [−2] . In fact the only obtainable textures if there are only these 3 flavon VEVs are the textures explored so far. {∆} ≥ 0 has no allowed couplings due to holomorphy, and for {∆} < −6 the only possibilities repeat the existing patterns as the matrix structure is driven only be the flavons contracting with Q i and L j . Even at higher order it is not possible with this field content to obtain linear combinations of the textures. The reason for this is clearer when considering a specific example, so we skip {∆} = −7 as it has no allowed couplings, and take {∆} = −8 where: is invariant, with the same texture as λ [−2] but with overall U(1) F charge +6 added due to SU(3) F singlet ( ijk φ i 3 φ j 23 φ k 123 ). Even though U(1) F would allow to swap flavon contractions to ∆(φ g and so on, all except eq. (4.5) automatically vanish due to the Levi-Civita tensor. See text  1  1  1  3  3  1  1  1  1  1  1  3  3 1 1  3  3 1

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The first A 4 framework we consider is of SUSY A 4 ×Z 3 ×U(1) FN 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) FN , generating the hierarchy in the charged lepton masses without requiring small Yukawa couplings; it can easily be used to justify the quark hierarchies as described in appendix B. SUSY's holomorphy together with the Z 3 separate the charged lepton sector and the neutrino sector. The A 4 triplet flavons φ l and φ ν acquire vacuum expectation values (VEVs) in special directions (1, 0, 0) and (1, 1, 1), respectively, shaping the leptonic mixing. The fields charged as 2 under the R-symmetry U(1) R are the alignment fields and are responsible for giving the flavons their VEV directions. We do not discuss the details of SUSY breaking, but when it occurs U(1) R is broken to leave only a Z 2R subgroup which distinguishes the SM fermions, i.e. it acts as R-parity. We refer to table 2 for details about the charge assignments. We have not listed the A 4 representation of ∆ in table 2 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 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. (2.1) -or, with a SU(2) triplet ∆, one would have λ ij [Q i ∆L j ] as a SUSY version of eq. (2.2). 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).

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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. (2.6), 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 µ generation was merely assumed.
This isolation of lepton generation takes place at LO in a generic expansion parameter of φ /Λ, where φ represents the flavons and Λ the scale of new physics associated with the breaking of A 4 . 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 θ 13 , which fixes the expansion parameter φ /Λ ∼ 0.2.
With a sizable expansion parameter it is important to consider contributions beyond LO, i.e. terms featuring additional flavon insertions (of the triplet flavons or of ξ , a 1 of A 4 ). For the {∆} = 2 case that produces lepton isolation, multiple insertions of φ l involve the combination [φ l φ l ] 3s which gives contributions to λ that can be reabsorbed into the LO one, due to its effective (1, 0, 0) VEV; on the other hand both φ ν and ξ carry non-trivial Z 3 charge so they can only appear as multiples of 3 so the earliest contribution appears at NNNLO which is of the order ( φ /Λ) 3 , hence sub-percent. We conclude that in this case, contributions beyond LO can only change the structure of λ negligibly, and lepton isolation holds to a very good approximation.
When considering effects beyond LO it is relevant to reconsider option 4., where {∆} = 0 leads to LO lepton universality due to the coupling with φ ν . The Z 3 always JHEP06(2015)072 See text  1  1  1  3  3  1  1  1  1  1  3  3 1 1 1  3  3   allows adding the neutral φ l to this scenario, so there are NLO contributions involving [φ l φ ν ] 3a,3s . The effective VEVs of the combinations are respectively φ 1 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 4.3.1.
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) FN nor θ are present, with the charged lepton hierarchy being due to a field θ . 6 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 . 7 When adding a leptoquark to this framework one can again obtain LO λ structures like eq. (2.6) by having the Z 4 charge of ∆ be {∆} = 2. At LO we have: 6 To avoid confusion with the ξ of section 4.2, we renamed this field to θ . In [10][11][12] the same field is named ξ . 7 In this A4 × Z4 framework, the presence of ξ allows viable θ13 for a region of parameter space that is not fine-tuned, as discussed in detail in [12].
They are similar to options 1.,2.,3. of section 4.2, 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 4.2, 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: 1. ∆ ∼ 1 has LO coupling to τ , φ i l L i ξ ∆ NLO coupling to electron, φ i l L i ξ ξ ∆ NNLO coupling to µ.
We illustrate these in matrix form, defining κ ≡ ξ /Λ ∼ 0.2: (4.8) where we also parametrized the quark suppression factors as in eq. (2.7) for a more direct comparison. The first pattern, λ [1] cannot simultaneously accommodate R K and B s -mixing constraints unless M few TeV. The second one, corresponding to an inverted hierarchy with leptons flavor ordering of leptoquark couplings in opposite way as the one to the Higgs, and the third one, normal hierarchy (for electrons) are both viable. Phenomenological implications are summarized in section 4.4.
To conclude, within A 4 × Z 4 it is possible to obtain special versions of eq. (2.7). Note also that, if instead of ξ we had considered a Z 4 neutral 1 flavon -which would be difficult to distinguish in terms of lepton mixing angles [12] -for the same LO contribution, replacing the 1" with the 1' leads to swapping which lepton generation is coupling at NLO and at NNLO, i.e, effectively swapping in each matrix in eq. (4.8) the κ with the κ 2 terms. The resulting λ structures are different and can be tested experimentally. As a corollary of that, by having simultaneously both 1 and 1 one can obtain LO in one lepton generation and at NLO the other two, e.g. LO coupling to µ, NLO to e and to τ . The variant with LO

Quarks non-trivial under A 4
An interesting option that can be considered is to make different generations of quarks transform as different non-trivial singlets of A 4 . Consider option 1., 2., 3. of the A 4 × Z 4 framework. If all quark generations are the same (non-trivial) A 4 singlet, it just shifts which lepton generation is isolated at LO by each leptoquark choice. For instance, if Q i are all 1 , then ∆ ∼ 1 is the leptoquark that would now couple to τ , whereas ∆ ∼ 1 which previously isolated τ would now couple instead to µ.
On the other hand new flavor pattern arise when different generations of quarks are transforming under different A 4 singlets. We note that some care is necessary as this possibility may lead to unwanted implications for the Yukawa couplings with the Higgs and a CKM matrix that is not viable. Indeed this is generally the case for the A 4 ×Z 3 framework.
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: 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. (4.10) are phenomenologically viable:  Eq. (2.7) with κ ∼ 1, i.e. eq. (4.7), λ NLO Eq. (2.7), normal hierarchy, i.e. eq. (4.8)λ [1 ] Eq. (2.7), inverted hierarchy, i.e. eq. (4.8)λ [1 ] Eq. (4.10) , eq. (4.11) two-flavor Table 4. Summary of viable λ structures and how they can be obtained through a flavor symmetry. In all cases except eq. (4.2) and eq. (4.10) the quarks transform trivially under the flavor symmetry.
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. 8 In fact, allowing only a minimal set of messengers we can obtain λ matrices like eq. (4.10) with x 1 = 0, corresponding to the truly minimal patterns from [3] that explain LNU data eq. (1.1).

Summary of flavor symmetry frameworks and predictions
We summarize in table 4 the features of the SU(3) F and A 4 frameworks discussed in the previous sections, for a single leptoquark.
The phenomenology of the patterns following from the hierarchy ansatz eq. (2.7) has been detailed in section 3. The two-lepton flavor patterns eq. (4.10) and eq. (4.11) discussed in section 4.3.1 that arose newly in the analysis of the flavor symmetries, have predictions similar to the single lepton flavor pattern eq. (2.6): they successfully explain R K by either BSM in b → se + e − or b → sµ + µ − transitions, however LFV signals can appear in addition in either eµ, eτ or µτ related to b → d or s → d transitions. For the example in eq. (4.10): B → πτ µ, B → τ µ and τ → K ( * ) µ, but none involving electrons. The sub-case for x 1 = 0, that is no couplings to d-quarks, can also be obtained, corresponding to the minimal benchmark patterns presented in [3], to which we refer for their phenomenology. They have no LFV. Implications include effects in b → (d, s)νν decays, relevant for the Belle II experiment. Correlations between B → Kνν and B → K * νν have been worked out in [41].
The other genuine non-Abelian type of λ structure is the inverted hierarchy eq. (4.8), 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 3 using κ = κ , κ = κ 2 and κ = κ 3 , respectively. The normal hierarchy pattern, λ [1 ] of eq. (4.8), predicts LFV in processes involving eµ, JHEP06(2015)072 eτ and µτ corresponding to κ = κ 2 , κ = κ 3 and κ = κ , respectively. In these cases, . 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.

Rare Higgs decays
In this section we discuss leptoquark effects in decays of the Higgs into fermion antifermion. 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.

JHEP06(2015)072
Spurion analysis shows when LFV in the Higgs coupling¯ L Xe R is induced, cf. eq. (2.1), 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. (2.6) 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. (5.4) is the same which drives → γ. We find that, using the limits from table 1, LFV Higgs effects are limited as y µτ y τ 10 −3 , y eτ y τ 10 −3 , y eµ y µ 10 −6 . (5.5) Flavor models of course predict patterns for these couplings and relate h → decays. E.g., in terms of the parametrization eq. (2.7): 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 4.4. The CMS data on the h → τ µ branching fraction [7] imply a sizable off-diagonal Yukawa-coupling to the SM-like Higgs |y τ µ | 2 + |y µτ | 2 = (2.6 ± 0.6) · 10 −3 , about a third of the coupling to the taus itself, y τ 0.01 tan β. Such a sizable effect, taken at face value, exceeds the leptoquark estimates eq. (5.5) and points to mechanisms beyond perturbative loops. An example is given by [44,45].

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. (1.1) 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 3 and 4.4. The LFV branching ratios, whose size is driven by (1 − R K ) 2 are in JHEP06(2015)072 part accessible to near-term future experiments, including MEG, LHCb and Belle II, the details of which depend on the specific flavor pattern of the leptoquark coupling λ, single lepton flavor eq. (2.6), the hierarchy pattern eq. (2.7), or further patterns which follow from the non-Abelian nature of the underlying flavor symmetry. Viable and realistic ones are summarized in table 4. They can be distinguished phenomenologically. Further data could not only be a discovery of BSM, but also provide clues about flavor. Already current data rules out many choices of flavor group assignments, as demonstrated in section 4.
Indeed if certain patterns of leptoquark coupling are observed in the future, they constitute strong hints of the presence of non-Abelian flavor symmetries. From the examples studied in this paper we single out three patterns. One is predicted in the unified SU(3) F framework, and is highly symmetrical with four entries of the same magnitude, eq. (4.2). Alternatively if lepton isolation, either in electrons on muons, eq. (2.6), 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 chirality-structure 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. (2.1) and eq. (2.2) can be distinguished. Pinning down the chirality of the leptons is possible by comparing the LFV branching ratios B (s) → + − with B (s) → − + , which pick up different lepton mass factors. This method of diagnosing quark and lepton chirality, of course, applies model-independently for the low energy theory eq. (A.1), i.e., is not limited to leptoquark mediated FCNCs.
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 pairproduced 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.

A Leptoquarks in b-decays
This appendix explains how to obtain predictions for b-decays from the leptoquark interactions eq. (2.1) and eq. (2.2), corresponding to ∆(3, 2) 1/6 and ∆(3, 3) −1/3 , respectively. After integrating-out and fierzing [31], b → s + − transitions with = and = are induced. Employing the common |∆B| = |∆S| = 1 effective Hamiltonian 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' fourmomentum, 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. (A.1), eq. (A.4) 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 3 fb −1 data by LHCb on B → K * µµ angular observables [49] taking into account SM uncertainties is beyond the scope of our work.

B Quarks and Froggatt-Nielsen
Relative suppression between rows of λ is naturally obtained by assigning different U(1) FN 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) FN charge of a field.
Choosing non-zero values of FN-charges for the leptoquark significantly impacts λ and the resulting phenomenology, which forces {∆} to be within 0 and ∼ 3: {∆} > 0 adds an overall suppression to all leptoquark couplings of {∆} , which, while keeping ρ d /ρ unchanged, leads to suppression of λ 0 . This requires lighter leptoquarks to explain R K . This case is discussed in scenario C of section 3. Too large charges {∆} 3 cannot accommodate simultaneously eq. (2.4) and direct search limits M 1 TeV. Within SUSY negative charges are in conflict with eq. (2.4), because due to holomorphy, {∆} < 0 forbids couplings to b-quarks. To see this take {∆} = −1 and compare to {Q 3 } or {d c 3 }, which are zero to account for the mass of the b and top quarks.

C A few A 4 details
A 4 has 4 distinct irreducible representations, three of them are 1-dimensional, i.e. singlets, and one of them is 3-dimensional, i.e. the A 4 triplet. The trivial singlet we label 1, and it transforms trivially under A 4 . The (non-trivial) singlets 1 and 1 are conjugate to one another and they transform under a specific A 4 generator, T , by getting multiplied respectively by ω 2 and ω (ω ≡ e i2π/3 , with ω 3 ≡ 1). The product of A 4 singlets has (1 × 1 ), (1 × 1 ), (1 × 1 ) transforming as 1 , 1 , and 1 respectively. The action of the group on triplets is represented by 3×3 matrices. Consider specific triplets A = (a 1 , a 2 , a 3 ), . Under generator T , which is diagonal in the basis we are considering, we have T A = (a 1 , ω 2 a 2 , ωa 3 ) (same for B). We use the conventions in [11,12], and square brackets to indicate A 4 products: It is also possible to construct a symmetric (s) and an anti-symmetric (a) triplet:

D (Anti-)triplet ∆
Within the frameworks we considered in section 4 it also possible to consider 3 generations of leptoquarks transforming as a representation of the flavor symmetry. This is more intricate than the single leptoquark scenarios we focused on, as there are three λ structures (one for each generation of the ∆ multiplet), and the flavor structure of the mass matrix for the ∆ generations is also relevant. In the frameworks considered here ∆ only acquire masses after the R-symmetry is broken (the leptoquark mass terms behave similarly to the µ-term µh u h d ). As we will show, in some cases there are holomorphic ∆ bi-linears that could couple to whatever superfield is responsible for breaking the R-symmetry.

D.1 SU(3) F
In the SU(3) F framework one can have anti-triplet ∆ i (i.e. 3 generations of leptoquark transforming like the φ i ) or instead a triplet ∆ i (i.e. 3 generations of leptoquark transforming like the SM fermions). Discussing full models is beyond the scope of the present paper, but we illustrate some structures that can arise. For triplet ∆ i , the invariants can be quite different from those discussed in section 4.1. 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 which occurs for {∆} = −3, where ∆ 1 decouples and ∆ 2,3 couple with λ 2,3 ∼ λ [−2] , eq. (4.2). For this charge assignment, there is a holomoprhic bi-linear (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
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 .

D.2 A 4
If we have 3 generations of leptoquarks as an A 4 triplet ∆ i there can be invariant contractions [L∆] = L 1 ∆ 1 + L 2 ∆ 3 + L 3 ∆ 2 , if under Z 3 , {∆} = 2; within A 4 × Z 4 the choice would be {∆} = 3. Each generation has its own λ i matrix. The LO-structures are: (D.5) 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.

JHEP06(2015)072 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(5). Viable down quark masses can be obtained with LO contractions to the φ l flavon, with NLO corrections from non-trivial singlet flavons and the up sector enabling viable CKM mixing.
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