Testing Minimal Flavor Violation in Leptoquark Models of the $R_{K^{(*)}}$ Anomaly

The $R_{K^{(*)}}$ anomaly can be explained by tree level exchange of leptoquarks. We study the consequences of subjecting these models to the principle of minimal flavor violation (MFV). We consider MFV in the linear regime, and take the charged lepton Yukawa matrix to be the only spurion that violates lepton flavor universality. We find that a combination of constraints from a variety of processes -- $b\to s\mu\mu$, $b\to s\tau\tau$, $b\to s\nu\nu$, $b\bar b\to\tau\tau$ and $b\to c\tau\nu$ -- excludes MFV in these models.


I. INTRODUCTION
Within the Standard Model (SM), lepton flavor universality (LFU) is respected by the weak interactions. Consequently, LFU is predicted to hold -up to (calculable) phase-space effects -in processes where the Yukawa interactions are negligible. Hints of violation of LFU have, however, been observed by the LHCb experiment in B → K ( * ) ℓ + ℓ − decays. While LFU implies that the ratios (1) (q 2 is the invariant dilepton mass-squared) should be very close to unity, the measurements give [1, 2] R K,[1,6]GeV 2 = 0.745 +0.090 −0.074 ± 0.036, R K * ,[1.1, 6.0]GeV 2 = 0.69 +0. 11 −0.07 ± 0.05, R K * ,[0.045,1.1]GeV 2 = 0.66 +0. 11 −0.07 ± 0.03, (2) which stand in a 2.2 − 2.6σ discrepancy with the SM predictions. The discrepancy, if not a statistical fluctuation, requires new degrees of freedom. In this work we focus on new physics models where heavy new bosons contribute to b → sµ + µ − transitions at tree level. Such new bosons can be SU (3) C singlets or triplets. We focus on the latter class, i.e. on leptoquark models [3]. More specifically, we consider simplified models where a single leptoquark representation is added to the SM fields.
There are eight leptoquark representations that have couplings to down-type quarks and to charged leptons. The R K ( * ) measurements suggest that the integration out of these leptoquarks generate an effective four-fermi operator of the form C bsµµ (s L γ µ b L )(µ L γ µ µ L ). ( Accordingly, the eight leptoquark representations can be divided to three groups: • One of the scalar leptoquark representations, couples down quarks to neutrinos and up quarks to the charged leptons, so it does not generate (at tree level) the operator of Eq. (3).
In this work, we thus focus on the three simplified models of Eq. (6). The requirement that the contribution of the leptoquarks to R K ( * ) breaks LFU implies that the leptoquark couplings have a non-trivial flavor structure. In particular, they must break the accidental SU (3) Q × SU (3) L global symmetry of the gauge interactions. Generic breaking would lead to unacceptably large contributions to various flavor changing processes. This situation is the specific realization of the new physics flavor puzzle [10] in the leptoquark framework. Thus, the R K ( * ) measurements provide an opportunity to test the various ideas that have been proposed to solve this puzzle [5,[11][12][13]. Arguably the simplest, and the most easily falsifiable of these is the principle of minimal flavor violation (MFV) [14]. In this work we ask whether the leptoquark models that explain the R K ( * ) anomaly can be MFV (see [5] for related work).
Within the MFV framework, various flavor changing processes are related to each other. For example, the b → sµ + µ − transition relevant to R K ( * ) is related to the b → sτ + τ − and the b → dµ + µ − transitions. We ask whether the MFV relations exclude some or all of the three otherwise-viable leptoquark models.
The plan of this paper goes as follows. In Section II we present the principles of applying MFV on leptoquark couplings. In Section III we obtain the viable lepton flavor representations for leptoquarks, and exclude some of the gauge representations that would be viable if MFV were not imposed. In Section IV we test the various quark flavor representations against experimental constraints. We present our conclusions in Section V. Several additional phenomenological constraints are discussed in Appendices: B s − B s mixing (Appendix A), direct LHC searches (Appendix B), perturbative unitarity (Appendix C), pp → µµ (Appendix D), and s → uτ ν (Appendix E).

II. MFV FOR LEPTOQUARKS
In this section we discuss in more detail the implementation of MFV in leptoquark models [15]. In the absence of Yukawa couplings, the SM acquires an accidental non-Abelian global symmetry, The Yukawa couplings, Thus, the three Yukawa matrices can be taken as spurions with the following transformation properties under G flavor : Imposing MFV on the SM extended with leptoquark fields means that we assign the leptoquark fields with well-defined transformation properties under G flavor and require the following: • All terms made of SM fields, leptoquark fields and the Yukawa spurions are formally invariant under G flavor .
One subtlety relates to the definition of minimal lepton flavor violation. We consider the case that the only spurion that breaks SU (3) 2 ℓ is Y E . If one takes into account the fact that neutrinos are massive, additional spurions may play a role. For example, if neutrino masses arise from a seesaw mechanism with three heavy SM-singlet fermions N , then G flavor is extended by an SU (3) N factor, and both M N , the mass matrix of these fermions, and Y N , the neutrino Yukawa matrix, break the flavor symmetry. Taking Y E to be the only leptonic spurion is equivalent to assuming that the seesaw scale is higher than the scale at which the leptoquark couplings are set. Moreover, if this scenario holds in Nature, it explains why lepton flavor violation (e.g., µ → eγ [16]) has not been observed except in neutrino oscillations.
We are interested in leptoquarks that generate the effective four-fermi operator of Eq. (3). Thus, the SU (3) C × SU (2) L × U (1) Y invariant operator must involve the leptoquark field, the quark doublet fields Q i and the lepton doublet fields L j . Since our starting point is the anomaly in b → sµ + µ − transitions, we work in the down and charged lepton mass basis. Hence the quark doublets are Q d,s,b and the lepton doublets are L e,µ,τ . In this basis, the three Yukawa spurions have the form To have a predictive framework for processes that involve the third generation fermions (in particular the b-quark and the τ -lepton), we make two assumptions: 1. The spurions related to Y D and Y E are small enough to keep the leptoquark couplings perturbative.
2. Terms that are higher power in Y F (F = U, D, E) are suppressed compared to lower powers.
The first assumption can be satisfied in the models that we consider for leptoquark masses not much heavier than a few TeV. A quantitative analysis is given in Appendix C.
The second assumption means that we do not consider MFV in the nonlinear regime [17]. The implications of relaxing this assumption are briefly discussed in Section V. The only case where we include spurions that are quadratic (or higher order) in the Yukawa couplings is when the leading contribution to flavor changing couplings arises from the operator In the down mass basis, and neglecting y c and y u , it has the form

III. MINIMAL LEPTON FLAVOR VIOLATION (MLFV)
For the sake of concreteness we continue by considering a specific model out of the three -that is the T (3, 3) −1/3 model -but at this stage the lessons drawn are common to all three. The leptoquark couplings of T have the form where ǫ = iτ 2 , and τ a are Pauli matrices in SU (2) L . Integrating out T , we obtain the following EFT Lagrangian: A. B → K ( * ) ℓ + ℓ − The relevant leptoquark models generate, among others, operators of the following form: with We consider the experimental data, BR(B + → K + τ + τ − ) < 2.25 × 10 −3 [18], and BR(B + → K + µ + µ − ) = (4.4 ± 0.3) × 10 −7 [19], which give We now examine various possibilities for the representation of T under SU (3) 2 ℓ and their predictions for λ αi and, consequently, for C bsµµ and C bsτ τ .
The spurion must transform as (1 + 8, 1) SU(3) 2 ℓ and thus Given the smallness of the lepton Yukawa couplings, we expect that the leading contribution is lepton-flavor universal and thus cannot account for the R K ( * ) anomaly. It could, however, be that the singlet contribution is negligibly small for some reason, and the octet contribution dominates. In the case of octet-spurion dominance, Taking into account that the O(0.25) deviation of R K from unity comes from the interference of the SM and leptoquark amplitudes, we find that Eq. (20) implies R τ /µ ∼ 10 8 , strongly violating the experimental upper bound of Eq. (17). We conclude that having a leptoquark transform as (3, 1) For all such models, we have the ratio between the T -mediated amplitudes given by Thus, these models predict a factor of 4 below the present bound.
In the previous subsection, we proved that the only viable lepton flavor representation is (1,3) SU(3) 2 ℓ . In this subsection we use the experimental data on B → K ( * ) νν to exclude some of these models.
Experiments put the upper bounds BR(B + → K + νν) < 1.6 × 10 −5 [20] and BR(B + → K * + νν) < 4.0 × 10 −5 [21]. Thus, The relevant leptoquark models generate, among others, operators of the following form: The SM predicts [22,23] (Note that C SM bsνν is the value for a single flavor, and thus the SM prediction is R ν/µ ∼ 6.6.) The R K ( * ) anomaly requires We now obtain the ratio C NP bsντ ντ /C NP bsµµ for each of T , U µ 3 and U µ 1 , and the resulting prediction for R ν/µ : Note that for R τ /µ we had to consider only the lepton flavor representation of the leptoquark. In contrast, for R ν/µ , the result depends also on the Lorentz and SU (2) × U (1) representation and is thus different among the three models.
• U µ 1 (3, 1) +2/3 : We conclude that, for the (1,3) SU(3) 2 ℓ representation, the T and U µ 3 models are excluded by the upper bound on R ν/µ . On the other hand, the U µ 1 models predict this ratio to be a factor of 4.7 below the present bound (or, equivalently, 1.3 above the SM prediction).

C. Summary of MLFV
There are four classes of MLFV models for leptoquarks that can a-priori (that is, without imposing MLFV) generate the operator of Eq. Only the latter class is good for explaining the R K ( * ) anomaly (without violating the R τ /µ bound). This MFV classification is common to all three viable leptoquark models: T (3, 3) −1/3 , U µ 3 (3, 3) +2/3 , and U µ 1 (3, 1) +2/3 . However, additional processes put further constraints: • The upper bounds on BR(B → K ( * ) νν) exclude the MFV-T and MFV-U µ 3 models. • In Appendix A we show that the MFV-T model is excluded also by the upper bound on new physics contribution to B s − B s mixing.
We conclude that the only model that is not excluded by the above consideration is the U µ 1 model in the (1,3) SU(3) 2 ℓ representation.
To make further progress, we need to consider the SU (3) 3 q representation of the leptoquark, which we do in the next section.

IV. MINIMAL QUARK FLAVOR VIOLATION (MQFV)
We now consider the possible SU (3) 3 q representations of the U µ 1 leptoquark. For simplicity, from here on we omit the sub-index 1 and the Lorentz super-index µ and denote the Lorentz-vector in the (3, 1) +2/3 gauge representation simply by U .
The reason is that no combination of Y U 's and Y D 's transforms as (3, 1, 1) SU(3) 3 q . We conclude that U must transform as a triplet under SU (3) 3 q and as an anti-triplet under SU (3) 2 ℓ . If indeed U transforms as a quark-flavor-triplet and lepton-flavor-anti-triplet, then there are nine U -flavor states, that can be denoted as The 9 × 9 mass-squared matrix of the U flavor states transforms as either 1 or 1 + 8 under each of the five SU (3)'s. Given the smallness of all Yukawa couplings except y t , and the smallness of |V ts | and |V td |, the 9 × 9 mass-squared matrix is near diagonal, so that we can call the nine U mass-eigenstates by the same names as the flavor states, namely U αi . Furthermore, given our assumption of small SU (3) 2 ℓ spurions, the masses are lepton-flavor universal to a good approximation. As concerns the quark-flavor, in some cases the b-states (t-states) are separated by O(y 2 t ) from the s-and d-states (u-and c-states), but in any case there is no hierarchy.
Given the couplings in Table I, we can now translate the R K ( * ) requirement, into a constraint on the model parameters.  (1, 3, 1, 1,3) bδityαytV * tj (UD) αi (1, 1, 3, 1,3 • U Q : In order that to have destructive interference with the SM amplitude, we need 2Re(x 1/8 ) + y 2 t |V tb | 2 > 0, namely (assuming that x 1/8 is real) • U D : In order that to have destructive interference with the SM amplitude, we need x 1/8 + y 2 t |V tb | 2 > 0, namely Within the MFV models that we study, the requirement that the leptoquarks contribute to the Wilson coefficient of the operator of Eq. (3), namely to the b → sµ + µ − decay, implies that they contribute also to bb → ℓ + ℓ − scattering processes [24]. MFV suggests that the largest contribution will be to the final τ + τ − state. This contribution is constrained by the LHC searches for the τ + τ − signature.
In Ref. [24], the results of the ATLAS searches [25,26] have been recast into bounds on vector leptoquarks mass and coupling: where the stronger (weaker) bound applies in case that M U > 2 TeV (M U ∼ < 2 TeV), which is above (within) the LHC direct reach. The bound for M U ∼ < 2 TeV is not constant and is slightly weaker than 4 TeV −2 below 1 TeV, which is anyway excluded by the LHC direct searches (see Appendix B). In Fig. 1 we present the excluded region for M U < 2 TeV, compared to the 1 σ allowed region to fit the R K anomaly.
Within our models, we have [see Eq. (A5)] C bsτ τ = 0.28 TeV −2 . We thus require The MFV prediction for C bbτ τ depends on the quark flavor representation: The 1σ allowed range from R K ( * ) , the region excluded by the high-p T pp → τ τ search [24], and the region excluded by LHC direct searches (see Appendix B).
• U U : which is excluded.
• U D : Eqs. (44) and (46) imply a narrow allowed window: We conclude that, within the MFV framework, the combined constraints from R K ( * ) and bb → τ + τ − exclude the U U and U Q scenarios, and leave the U D model as the only viable one.
Within the MFV framework, the leptoquarks that generate the effective term of Eq. (3), relevant to b → sµµ decays, generate also the term In contrast to the b → sµµ and other processes discussed so far, the b → cτ ν τ decay is a quark-flavor changing charged current process. We have The data require C NP bcτ ν ∼ 0.17 TeV −2 . Together with the R K ( * ) constraint, we need In the U D model, we have the following prediction for C bcτ ν : Thus, where the range corresponds to −1 < x 1/8 ∼ < −0.4. The U D model predicts a strong suppression (by at least 10%) of R D ( * ) from the SM prediction and is thus excluded. In fact, it will remain excluded even if R D ( * ) turns out to be consistent with the SM prediction (with experimental uncertainties no larger than the present ones) as long as R K ( * ) is substantially suppressed compared to the SM.
We discuss additional aspects of R D ( * ) within the MFV framework, independent of R K ( * ) , in Appendix E.

D. Summary of MQFV
There are three classes of MQFV for leptoquarks that can generate the operator of Eq. (3): • The U U model in the (1, 3, 1) SU(3) 3 q representation. It is excluded by a combination of the R K ( * ) and bb → τ + τ − measurements.
• The U Q model in the (3, 1, 1) SU(3) 3 q representation. It is excluded by a combination of the R K ( * ) and bb → τ + τ − measurements.
We conclude that all MFV models considered by us are excluded.

V. CONCLUSIONS
The R K ( * ) anomaly can be accounted for in models where there is a significant contribution to the b → sµ + µ − transition from the tree level exchange of leptoquarks. The pattern of deviations from lepton flavor universality (LFU) allows three simplified models, each with a single new leptoquark field:  , (3, 1, 1), (1, 3, 1) or (1, 1, 3).
MFV relates the measured B → K ( * ) µ + µ − rates to various other processes, such as B → K ( * ) νν, bb → τ + τ − and b → cτ ν. We summarize our use of these relations to test MFV in Table II. Additional measurements (B s −B s mixing, direct leptoquark searches, pp → µ + µ − , τ → sūν) and considerations (perturbative unitarity) which are relevant to leptoquark models that aim to explain the R K ( * ) anomaly, are discussed in Appendices.
TABLE II: MFV-predictions of simplified leptoquark models that account for the R K ( * ) anomaly. R ν/µ is discussed in Section III B, Γ bb→τ τ in Section IV B, and R D ( * ) in Section IV C. A super-index * means that consistency with the observable applies for a small range of the parameter x 1/8 .
Before we state our conclusions, let us repeat the ingredients of the models that we consider: 1. Simplified models, with a single leptoquark representation; 2. The leptoquark contribution to the b → sµ + µ − transition occurs at tree level; 3. The only spurion that breaks the lepton flavor symmetry is the charged lepton Yukawa matrix.
4. MFV is in the linear regime (higher powers in the spurions are suppressed compared to lower ones).
Most of our conclusions hold, however, in generic such extensions of our framework. For example, even with tree level contribution to B → K ( * ) µµ, the MFV framework predicts that the third generation couplings of the leptoquarks are close to the perturbative limit. If the contribution is suppressed by an additional loop factor, then these couplings will be pushed to non-perturbative values. As another example, if we allow neutrino-related spurions to play a significant role in lepton flavor conserving processes, it will be hard to avoid too large contributions to lepton flavor changing ones, such as µ → eγ [44]. Our conclusions do not hold, however, if MLFV is in the nonlinear regime. In this case, the strict relations between τ and µ couplings do not hold. Specifically, the bounds from R τ /µ , R ν/µ , bb → τ + τ − and B s − B s mixing cannot be strictly applied. Yet, for some of the constraints, fine-tuned cancelations between the linear term and the higher order ones are needed to satisfy the constraints, which goes against the spirit of MFV. Order one modifications of the linear MFV prediction can, however, bring U µ 1 models into consistency with the Γ bb→τ τ and R D * constraints. In fact, the phenomenology of models of nonlinear minimal flavor violation [17] is similar to that of U (2) models, which have been shown to be viable candidates to explain the R K ( * ) anomaly [12] We find that all models are excluded by a combination of b → sµ + µ − , b → sτ + τ − and the processes presented in the Table. Note that for vector-leptoquark models, constraints from loop diagrams are sensitive to the UV completion of the model. It is thus important that we exclude these models based on tree level processes alone. (In Appendix A we consider a loop process, B s − B s mixing, but we confront it with only scalar leptoquark models. ) We conclude that if the R K ( * ) anomaly is experimentally established, then minimal flavor violation in the linear regime will be excluded. (A1) Fitting the mixing amplitude to the experimental ranges of ∆m Bs , ∆Γ Bs and a s SL , gives [45] Requiring that the contribution from the scalar leptoquark T is within ||∆ s | − 1| ∼ < 0.25 gives [46]: where we used ∆m Bs = 1.17 × 10 −11 GeV, m Bs = 5.37 GeV and f Bs ∼ 0.23 GeV. The R K ( * ) anomaly requires (see e.g. [47,48]) In the viable models, λ * τ b λ τ s = (y τ /y µ ) 2 λ * µb λ µs , so that Eqs. (A3) and (A5) can be simultaneously satisfied only for M T ∼ < 0.5 TeV. The members of the third generation T -triplet of charges −4/3, −1/3 and +2/3 decay into, respectively, τ b, νt and τ t. The latter has branching ratio 1 which leads to an exclusion of 850 GeV [49]. The recast [50] of the SUSY CMS analysis [51] for the ttνν topology leads to an even stronger bound of 1.07 TeV. Thus, M T ∼ < 0.5 TeV, as required by the ∆m Bs constraint, is excluded by LHC direct searches. We conclude that ∆m Bs constraints exclude the MFV-T model as a possible explanation of the R K ( * ) anomaly.
As concerns the case of vector leptoquarks, their contribution to B s − B s mixing is divergent. The divergence comes from the k µ k ν term in their propagator, i[(k µ k ν )/M 2 Ref. [46] suggests that a conservative bound can be obtained by considering the contribution of the g µν term only. The numerical factor of the mixing amplitude is four times larger than in the scalar case, and the resulting bound on the mass is therefore two times stronger, M U ∼ < 0.25 TeV, which is excluded by the direct searches. Yet, this bound is model dependent. To relax the bound by a factor of O(7) (see Appendix C), the contributions from the terms that we omitted should cancel with those that we took into account to the two percent level. The production cross section σ(pp → U µ 1 U µ † 1 ) for vector leptoquarks is considered in [47]. As already discussed in Section IV, MFV implies that the nine flavor states are almost degenerate. This represents then an important and distinctive feature of our framework.
Each of the nine flavor-states has a branching ratio of 50% to decay into a specific charged lepton and a jet, and a branching ratio of 50% to decay into a neutrino plus jet. For each final state topology searched for at the LHC, we define with We ignore the mixed final states In almost all cases, the decay is prompt. The only possible exceptions are the decays of U eq , with q = u or d, where the decay might have a displaced vertex.
The strongest bound come from the U µ 1 → eb search [52]: The U τ b state is the one related to both the perturbative unitarity bound, discussed in Appendix C, and to the pp → τ τ bound, discussed in Section IV B. It decays with branching ratios of 50% into τ b and 50% into νt, thus leading to 25% of the events with bbτ τ final state and 25% of the events withttνν final state. These final states are constrained by, respectively, the search for third generation leptoquarks and a recast [50] of the CMS SUSY search [51]: (The latter bound is significantly stronger than the reach of the dedicated leptoquark search for the final state ttνν [53] where the current limit is below 1 TeV.)
Requiring that the R K ( * ) anomaly is accounted for by U µ 1 gives Eq. (39). MFV relates i λ * µµib λ µµis to |λ τ τ bb | 2 . Consequently, the combination of Eqs. (C1) and (39) leads to an upper bound on the leptoquark mass, in particular on M τ b , which can then be compared to the direct lower bound presented in Eq. (B3) or, allowing for mass splitting within the U µ 1 multiplet, Eq. (B4).
which is excluded by Eq. (B3) but not by Eq. (B4). Thus, a mass splitting larger than 200 GeV between U τ b and U eb would be required to avoid the direct bounds and fulfill perturbative unitarity.
We conclude that, assuming quasi-degeneracy within the U µ 1 multiplet, the combination of perturbative unitarity and LHC direct searches excludes the U U and U Q flavor models.
Appendix D: pp → µ + µ − MFV models that contribute to C bsµµ of Eq. (3), generate also the terms C ssµµ (s L γ µ s L )(µ L γ µ µ L ) + C ddµµ (d L γ µ d L )(µ L γ µ µ L ). (D1) These terms contribute to pp → µ + µ − . Ref. [54] obtains from the experimental measurements the following bounds, which hold for leptoquark heavy enough that its effect on pp → µ + µ − is captured by EFT: Together with the R K ( * ) constraints, these bounds imply • U U : The U U contribution to pp → µµ is negligibly small.
• U D : Thus U D contribution to pp → µµ is negligibly small, except for a small region which is excluded, −1 < x 1/8 ∼ < −0.99.
where C suℓν is the Wilson coefficient of the term C suℓν (u L γ µ s L )(ℓ L γ µ ν ℓL ). (E3) The MFV models that we discuss contribute mainly to C suτ ν : We now obtain the predictions of the three U µ 1 MFV-models for C suτ ν and, where relevant, for C bcτ ν . • U U : The U U model predicts Thus, the U U model cannot account for the R D ( * ) anomaly independently of the R K ( * ) anomaly.
• U Q : and the U Q model predicts which is strongly excluded.
• U D : which is negligibly small. The U D model predicts The combination of the R D ( * ) and R K ( * ) constraints on U D was discussed in Section IV C. An analysis of the U D model with regard to R D ( * ) , independently of R K ( * ) , was carried out in Ref. [56].