Lepton flavor (universality) violation in rare kaon decays

Recent anomalies in the decays of $B$ mesons and the Higgs boson provide hints towards lepton flavor (universality) violating physics beyond the Standard Model. We observe that four-fermion operators which can explain the $B$-physics anomalies have corresponding analogs in the kaon sector, and we analyze their impact on $K\to\pi\ell \ell'$ and $K\to\ell \ell'$ decays $(\ell=\mu,e)$. For these processes, we note the corresponding physics opportunities at the NA62 experiment. In particular, assuming minimal flavor violation, we comment on the required improvements in sensitivity necessary to test the $B$-physics anomalies in the kaon sector.


I. PUZZLES IN THE FLAVOR SECTOR
The discovery of a Higgs-like resonance at the LHC experiments [1,2] provides the final ingredient to complete the Standard Model (SM) of particle physics. However, there are a variety of theoretical and phenomenological reasons to suspect that the SM is not the final theory, and that some form of new physics (NP) may also be present near the electroweak scale. While no direct evidence for physics beyond the SM was found during the first LHC run, there are some interesting indirect hints for NP in the flavor sector, chiefly in the semileptonic decays of B mesons and the SM-forbidden decay h → µτ of the Higgs boson.
In light of these flavor anomalies, we are prompted to consider possible effects of LFUV and LFV in rare kaon decays. One reason to expect correlations between the B meson and kaon sectors concerns the direct CPviolating ratio ǫ ′ /ǫ. Recent calculations in the large-N c limit [61,62] and on the lattice [63] suggest that the SM prediction for this quantity falls 2-3σ below the experimental world average ǫ ′ /ǫ = (16.6 ± 2.3) × 10 −4 [64][65][66].
However, the SM prediction for ǫ ′ /ǫ is sensitive to effects from ππ rescattering in the final state, which are entirely absent in the strict large-N c limit, while the lattice prediction [63] for the I = 0 phase shift δ 0 = 23.8(4.9)(1.2) • is about 3σ smaller than the value obtained in dispersive treatments [67][68][69]. Indeed, combining large-N c methods with chiral loop corrections can bring the value of ǫ ′ /ǫ in agreement with experiment [70,71].
Nevertheless, if the issue of final-state interactions is resolved in the future and the discrepancy persists, then NP contributions due to Z ′ bosons [72,73] or leptoquarks would provide a natural explanation. In that case, the B meson anomalies and tension in ǫ ′ /ǫ could originate from the same NP, with effects of LFUV and LFV in kaon decays to be expected. In the following, we do not commit ourselves to a specific NP model, but instead focus on the analogous four-fermion operators in the kaon sector which can give the required effect in semileptonic B decays.
For LFUV, the most natural processes to study are K → πℓ + ℓ − decays since these yield analogous observables to (1). However, we also consider the purely leptonic decays K → ℓ + ℓ − since the electron modes are within experimental reach (unlike B → e + e − ), and thus these processes are promising probes of NP operators which mediate LFUV. Limits on LFV can be extracted from K decays with µe final states.
branching fraction Br[K L → e + e − ] = 9 +6 −4 × 10 −12 [87]. For later use, these results are conveniently expressed in terms of the ratios which gives [81] R exp µµ = (1.25 ± 0.02) × 10 −5 , We do not consider the related K S → ℓ + ℓ − decays, since the SM predictions [88] lie well below the current experimental bounds [81]. The current limits on the LFV modes are listed in Table I. For the charged K decays, the sensitivity to LFUV and LFV is expected to improve at the high-statistics NA62 experiment [94][95][96], where the nominal number of decays is approximately a factor of 50 larger than that of NA48/2. 2 For example, the projected limit for Br[K + → π + µ + e − ] becomes 0.7 × 10 −12 . For K L decays, the KOTO experiment at J-PARC [98,99] has good prospects of reaching SM sensitivity for K L → π 0 νν. In principle, the increased reach might be sufficient to probe the K L modes involving charged lepton pairs, but the detection of these final states would require a different search strategy to the one employed for K L → π 0 νν. Finally, although we restrict our focus to the neutralcurrent sector, there is also renewed interest in chargedcurrent processes at the J-PARC E36 experiment, which is searching for signs of LFUV in K + → ℓ + ν ℓ [100].
On the theory side, all K decays have been studied thoroughly in the context of chiral SU (3) L × SU (3) R perturbation theory (χPT 3 ) [101][102][103][104][105][106][107][108][109][110][111][112], with the present status reviewed in [113]. The general picture that arises is the presence of long-distance physics, parametrized in terms of low-energy constants (LECs) in the effective weak Lagrangian. The values of these LECs are poorly known in most cases, and this limits the predictive power of χPT 3 in the weak sector. However, information on short-distance physics can be extracted by considering decay spectra as well as interrelations among different decay modes. Furthermore, LFV decay channels are typically less affected by hadronic uncertainties, and have been used in the past to extract limits on the NP scale [114]. Recently, the prospects of calculating the long-distance contributions on the lattice have been discussed [115], although it will take several years before high precision is reached.
This article is organized as follows. In Sec. II we establish our conventions and the general formalism necessary to study leptonic and semileptonic K decays. LFUV in K → πℓ + ℓ − decays is analyzed in Sec. III, where the assumption of minimal flavor violation (MFV) [116][117][118][119][120] is used to relate experimental limits in K and B decays. LFUV in the purely leptonic modes is discussed in Sec. IV, while the LFV decays are discussed in Sec. V. We conclude in Sec. VI.

II. FORMALISM
We follow the notation and conventions from [113]. To leading order in m −2 W and inverse heavy quark masses, the |∆S| = 1 interactions are defined by the effective Lagrangian where {Q i } is a set of local composite operators with Wilson coefficients C i . For the rare K decays under consideration, the relevant energy scale is µ ≪ m t,c,b , so we only need the four-quark operators Q 1-6 as well as the Gilman-Wise operators [121][122][123][124][125] We use α, β to denote color indices; otherwise the Dirac bilinearsf Γf are understood to be color singlets. For the Wilson coefficients we adopt the standard decomposition which arises from first decoupling t, W, Z simultaneously at µ = m W , followed by successively integrating out the b and c quarks in the evolution from µ = m W to µ < ∼ m c [126]. At zeroth order in the strong interactions and to O(g 2 ) in the weak interactions, C 2 is the only nonvanishing Wilson coefficient. At O(e 2 ), the γ, Z-penguin and W -box graphs in Fig. 1 generate nonzero coefficients for Q 7V and Q 7A [122], while O(g 2 s ) corrections generate nonzero contributions to C 1-6 .
Note that we have assumed right-handed quark currents are absent, as in the SM. This is because symmetrybased solutions to the anomalies in semileptonic B decays include 1) a left-handedsb current and a vectorial muon current, and 2) a left-handedsb current and a left-handed muon current. This pattern suggests NP effects involving similar operators in kaon decays.
The calculation of K → πℓℓ ′ and K → ℓℓ ′ amplitudes involves hadronic matrix elements such as γ * π|L eff |K , whose determination requires nonperturbative methods. These matrix elements can be systematically analyzed in χPT 3 , where amplitudes are expanded in powers of O(M K ) momenta p and quark masses m u,d,s = O(M 2 K ) (with m u,d /m s held fixed). For |∆S| = 1 transitions, the content of these calculations is summarized by an effective weak Lagrangian, constrained by the requirements of approximate chiral SU (3) L × SU (3) R symmetry and a discrete CP S symmetry [127], which interchanges the s and d quarks. The result is a set of effective weak operators which transform in the same way as L eff , i.e. in the (8 L , 1 R ) and (27 L , 1 R ) representations of the chiral group.
Empirically, it is observed that ∆I = 1/2 transitions dominate nonleptonic processes, which in χPT 3 corresponds to dominance by octet operators. It is not clear how this fact should be accounted for, although explanations based on large-N c [128][129][130][131] or an infrared fixed point in the three-flavor strong coupling [132,133] have been proposed. 3 In the context of potential NP contributions to C 7V and C 7A , one needs the chiral realization of the octet quark operator. At lowest order in χPT 3 , this is obtained by projecting the usual SU Here, U = U (π, K, η) is a chiral SU (3) field, and F 0 is the meson decay constant in the chiral limit, whose value can be determined from either the pion or the kaon channel. (Numerically, we use F π = 92.2 MeV and F K /F π = 1.22 [81].) For later convenience we also quote the analogous conventions for B decays [15] where At low energies, the dominant CP -conserving contribution to is known [101] to arise from single virtual-photon ex- where λ denotes the polarization of the photon. Barring the ∆I = 1/2 rule, there is no rigorous theoretical argument why (17) should dominate; after all, there are 3 A direct determination of the K → ππ amplitudes is not sufficient to explain the ∆I = 1/2 rule, since one cannot disentangle contact terms from effects due to final-state rescattering. Recently, a proposal [134] to separate these two contributions has been presented, based on a lattice measurement of K → π onshell. 4 Note that the relation (13) only relies on chiral symmetry. Large-Nc arguments [102,135] are needed only if a relation between the Gilman-Wise operators and corresponding LECs of the effective weak Lagrangian is sought. 5 In K L → π 0 ℓ + ℓ − decays, this contribution is CP -violating; see Sec. III C.  short-distance contributions from Z-penguin and W -box diagrams (Fig. 1). Moreover, it is not possible to make a clean theoretical prediction for the γ-penguin contribution C γ 7V associated with C 7V . As noted in [123,124], the QCD corrections to C γ 7V for t and c quarks are large and change both the magnitude and sign of the Wilson coefficient. Nevertheless, a rough estimate of the rate K → πℓ + ℓ − due to an amplitude ∼ C 7V gives a result far too small to explain the data. It is on this basis that short-distance contributions from Q 7V (as well as Q 7A ) are typically neglected in calculations of the branching ratios and spectra.
The requirements of chiral symmetry and gauge invariance imply that V + (z) vanishes at O(p 2 ) in χPT 3 [101], so the lowest-order contribution occurs at O(p 4 ). Beyond O(p 4 ), ππ rescattering in the nonleptonic decay K → πππ needs to be taken into account as well [104]. Given the limited information on most of the LECs, it is convenient to adopt a general representation [104] of the form factor which is valid at O(p 6 ). Here a + and b + parametrize the polynomial part, while the rescattering contribution V ππ + can be determined from fits to K → ππ and K → πππ data [136,137]. In general, V + receives contributions from both the octet and 27-plet parts of L eff [112], although the ∆I = 1/2 rule implies octet dominance, and thus the latter contributions are generally suppressed. The representation (20) was used as a fit function in all available high-statistic experiments with the results given in Table II. If LFU holds, the coefficients have to be equal for the electron and muon channels, which within errors is indeed the case. 6 Any discrepancy can then be attributed to NP, and thus the corresponding effect would be necessarily short-distance. It follows that the O(p 2 ) chiral realization (13) of the Q 7V operator converts the allowed range in a NP + into a corresponding range in the Wilson coefficients [102] and thus the difference between the two channels is If we assume MFV (to be understood in its simplest form, i.e. as the first order in the expansion in [119]), this translates into a constraint on the NP contribution to C B 9 : where we have averaged over the two electron experiments, defined λ t = V * ts V td , and used PDG global-fit values for the CKM matrix elements [81]. 7 In particular, we may use the modulus of λ t in (23) since MFV implies that the respective phases coincide with the SM, so that

10
(the remaining factors are simply due to the different normalizations of the effective Hamiltonians). Evidently, the determination of a µµ + − a ee + would need to be improved by at least an order of magnitude to probe the parameter space relevant for the B anomalies [15], whose explanation involves Wilson coefficients C B 9,10 = O(1). Progress in this direction can be anticipated at NA62, especially for the experimentally cleaner dimuon mode which currently has the larger uncertainty. It should be stressed that if NP does not satisfy MFV, the relative size of NP contributions to the Wilson coefficients is not fixed. In this case it is possible that the relative NP effects in the kaon sector are larger than in the B meson decays because the short-distance SM contribution is CKM suppressed in the former.
An alternative analysis strategy, often applied in B decays, to minimize sensitivity to hadronic form factors [13] relies on the ratio of branching fractions 6 Although note a small 1.6σ tension in the b + coefficient between the two electron experiments. We define LFU in the usual sense, i.e. excluding the Yukawa interactions in the SM (otherwise the different lepton masses would break LFU trivially). 7 In the estimate (23) we did not include effects due to renormalization group running between the scales of B-physics and χPT 3 . However, the semileptonic operators involve a vector or axial-vector current, so they are not renormalized (at the oneloop level). There is only a mixing of four-quark operators into the semileptonic operators, which is LFU conserving. where z min is a cutoff on the spectrum. While the impact of the muon mass is negligible in the B-physics case, this is not true for kaons and a lower z min must be applied to reduce the theory uncertainties. Indeed, as shown in Fig. 2, for given ranges in a + and b + the uncertainty in the ratio decreases quickly with increasing z min . However, in practice the determination of the ranges in the coefficients still requires a fit to the spectrum, so that all information on LFU can equivalently be extracted from this fit. It has been observed in [104] that the long-distance contributions could also be eliminated in the CPviolating charge asymmetry which in the SM is determined by Im λ t . Taking Im λ t = 1.35 × 10 −4 , the resulting SM prediction for (25) is ∼ 10 −5 [104]. This is to be compared with the most stringent experimental constraints A ee CP = (−2.2 ± 1.6) × 10 −2 [79], and A µµ CP = (1.1 ± 2.3) × 10 −2 [80], so we conclude that reaching SM sensitivity would require an improvement by 3 orders of magnitude.
In principle, there are additional axial-vector contributions to K + → π + ℓ + ℓ − , e.g. due to Z exchange ( Fig.  1) or NP mediators like Z ′ bosons or leptoquarks. This contribution generates an amplitude of the form where by analogy with (20), we take the lowest order decomposition A + (z) = d + for the axial-vector form factor.

Redoing the fit in terms of
gives d ee + = 0.00 ± 0.47 and d µµ + = 0.00 ± 0.13, which in turn yields the very weak bound |C B,µµ 10 − C B,ee 10 | < ∼ 1000. One critical factor in improving the accuracy of (23) concerns radiative corrections, which in [78][79][80] were performed according to the leading Coulomb factor [138,139]. More recently, these corrections have been addressed in full detail in a χPT 3 calculation assuming a linear form factor [110], in particular demonstrating that the corrections to the decay spectrum can still be expressed in a factorized form. These results should be valuable in view of the expected increase in statistics in the NA62 experiment.
While the extraction of short-distance physics from K + → π + ℓ + ℓ − decays themselves is difficult, a more precise measurement of its decay spectrum would have indirect implications for K S,L → π 0 ℓ + ℓ − : the numerical value of b + is larger than expected from dimensional counting or vector meson dominance (VMD), where the latter predicts b + /a + = 1/r 2 With increased statistics one might become sensitive to a quadratic term ∼ c + z 2 in the expansion of the form factor (20), and thereby test the hypothesis that VMD ought to be a decent description of V + once a non-VMD portion in a + related to sizable pion-loop contributions in this channel is subtracted [104,107]. Arguments along these lines are used to justify VMD assumptions in K S → π 0 ℓ + ℓ − , and, thereby, help fix the relative sign of the interference term between direct and indirect CP -violating contributions in K L → π 0 ℓ + ℓ − [107].
The expression for the K S → π 0 ℓ + ℓ − spectrum is very similar to (19), with neutral particle masses in the phase space expression and parameters a + , b + replaced by a S , b S in the form factor. Since the nonleptonic mode K S → ππ dominates the total K S width, the branching fraction for K S → π 0 ℓ + ℓ − is smaller than for the charged decay, and it is even more difficult to directly extract information on short-distance physics. However, a measurement of the spectrum would enable an explicit test of the VMD assumption for b S /a S = 1/r 2 V , which is expected to work better than in the charged channel due to the lesser role of pion loops. Use of the VMD assumption and the decay rates (5) implies that a S is only known with large uncertainties [113]: As we discuss in the next subsection, any additional information on K S → π 0 ℓ + ℓ − would sharpen the prediction of the indirect CP -violating contribution to K L → π 0 ℓ + ℓ − .
The process K L → π 0 ℓ + ℓ − is driven by three different decay mechanisms: a direct CP -violating 8 amplitude of short-distance origin from Q 7V and Q 7A , an indirect CP -violating transition due to K 0 -K 0 oscillations, and a CP -conserving contribution originating from K L → π 0 γγ and subsequent γγ → ℓ + ℓ − rescattering (with J = 0, 2, . . . two-photon states). The corresponding form of the decay spectrum as well as the consequences for extracting short-distance physics have been investigated in detail in [102,103,107,109,126,140]; here we review the salient features. First, the decay spectrum for the CP -violating part takes the form which in the limit of a purely vector interaction reduces to the neutral-channel analog of (19). The vector, axialvector, and pseudoscalar amplitudes are defined as Indirect CP violation leads to a vector amplitude of the form where ǫ ∼ e iπ/4 |ǫ| parametrizes K 0 -K 0 mixing, the ππ rescattering corrections have been neglected, and the second relation follows if VMD is assumed for the polynomial part. Short-distance physics only affects the direct CPviolating contributions with a vector or axial-vector lepton pair is CP -violating [113].
with Wilson coefficients as defined in (12) and K ℓ3 form factors f Kπ ± (z). Using the form-factor normalization f + (0) from [141], the slopes from [142], y 7V,7A from [107], and PDG input for the remaining parameters, we obtain for the decay rates More precise information on K S → π 0 ℓ + ℓ − would be highly beneficial for several reasons all related to the indirect CP -violating part of (33): its derivation relies on the VMD assumption for b S . As it stands, the dominant uncertainty resides in a S and the arguments put forward in [107] in favor of a positive sign of the interference term rely on the separation of VMD and non-VMD contributions to the polynomial coefficients, assumptions that could be tested with more precise data on K S → π 0 ℓ + ℓ − (and also K ± → π ± ℓ + ℓ − ). The CP -conserving contribution to the muon channel has been estimated to be [109] Br which is of the same order of magnitude as the CPviolating part. The CP -conserving electron decay channel is further suppressed [102,109,113]. Comparing (9)-(15), MFV suggests the identification y 7V,7A ∼ C B 9,10 α/2π, so that a NP contribution to C B 9,10 = O(1) would imply y 7V,7A = O(10 −3 ), about a factor of 5 less than the SM values of y 7V,7A . For a S = 1, the CP -violating branching fractions become Br[K L → π 0 e + e − ]| CPV = 2.8 × 10 −11 , Starting from this benchmark point, shifts in y 7V by ±10 −3 with y 7A held fixed (and vice versa) produce effects in the windows [2.5, 3.0] × 10 −11 and [6.9, 8.0] × 10 −12 , respectively, which in the case of the muon channel is even less than the uncertainty in the CP -conserving contribution (34). If NP were to obey MFV, a test of the B-physics anomalies in K L → π 0 ℓ + ℓ − therefore appears very challenging.
In Sec. III A we saw that the K → πℓ + ℓ − decays provided a probe of LFUV in NP scenarios involving vectorcurrent interactions. Here we examine the complementary role provided by K L → ℓ + ℓ − in constraining NP effects due to axial-vector interactions. 9 In these decays, there are both long-and short-distance contributions, with the former dominated by K L → γ * γ * → ℓ + ℓ − . As a result, it is convenient to normalize Γ(K L → ℓ + ℓ − ) to the K L → γγ rate (7), which can be expressed as where β ℓ = 1 − 4r 2 ℓ and the absorptive and dispersive components are [105,106,108,143,144] and The contact term χ(µ) arises from the counterterm Lagrangian [105,106,145] where Q = diag(2/3, −1/3, −1/3) is the charge matrix and χ(µ) = −(χ r 1 (µ) + χ r 2 (µ) + 14)/4 collects the finite parts χ r i of the LECs. It is conventional to decompose χ into long-and short-distance parts where the scale dependence of χ γγ (µ) compensates that from the term ∼ log m ℓ /µ. Although the SM prediction for χ SD is known, χ γγ depends on χ 1,2 whose values are not fixed by chiral symmetry. However, we can argue as before and observe that if LFU holds, then the SM values of χ must be equal in both the electron and muon channels. Then, using the chiral realization (13) of the V − A current, one obtains an analogous relation to (21) for the NP Wilson coefficient Channel χ (Solution 1) χ (Solution 2)  where we have defined Γ γγ = Γ(K L → γγ), N K = G F V ud V * us , and identified F 0 with the kaon decay constant F K . This implies that and thus NP limits can be inferred from precise extractions of χ in each lepton channel. Note that although χ is scale dependent, this dependence drops out in the difference (42). From the measured rates (8) one can use (36) and (37) to extract χ, up to a twofold ambiguity. The resulting values for each solution are shown in Table III, where we see that solution 2 for the electron channel is clearly ruled out. However, the present data are not precise enough to distinguish among the remaining solutions. The derivation of (42) relies on χPT 3 , generalized to include effects due to η-η ′ mixing. The leading contribution to the decay is mediated by pseudoscalar poles, P = π 0 , η, η ′ , and a constant form factor for the P → γ * γ * transition. At one-loop order, the P → ℓ + ℓ − decays all involve the same combination of LECs χ 1,2 introduced in (39) for K L → ℓ + ℓ − . In [146,147] the corresponding π 0 → e + e − amplitude was calculated, including full radiative corrections. Compared to Table III, the resulting extraction χ(M ρ ) = 4.5 ± 1.0 from the KTeV measurement [148] would favor solution 1 also for the muon mode. Moreover, the estimate for two-loop effects based on the double logarithm [147] χ LL (M ρ ) = 1 36 (43) indicates that at least for the pion-pole contribution to K L → ℓ + ℓ − , the one-loop formula should be sufficient. However, a similar estimate cannot be derived for the η channel since at two-loop order, SU (3) breaking effects render the decay amplitude sensitive to χ 1 − χ 2 as well. An explicit calculation [149] for η, η ′ → ℓ + ℓ − based on Canterbury approximants suggests that for those channels, LFUV two-loop effects are indeed significant. The potential impact of two-loop corrections has been investigated before in the context of K L → µ + µ − in [108,144], where large-N c and chiral arguments suggest that one can replace the (normalized) point-like form factor by the following parametrization f (q 2 1 , q 2 2 ) = 1 +α whereα is a free parameter. Based on this parametrization, the m ℓ -dependent terms in the γγ integral produce a shift in χ of the form [108] ∆χ(M ρ ) =α 3r 2 which yields ∆χ µµ − ∆χ ee = −2.8, where we have usedα = −1.69 as extracted from the slope in K L → ℓ + ℓ − γ [113]. Comparing to the numbers in Table III, we conclude that once the ee channel can be improved accordingly, additional input from phenomenology, K L → ℓ + ℓ − γ and K L → ℓ + ℓ − ℓ ′+ ℓ ′− , will be required to subtract the two-loop corrections and thereby identify potential LFUV contributions.
To illustrate the improvement required in the ee mode for such a test of LFUV in the interesting parameter space, we return to the one-loop relation (42) and again invoke MFV as in (23) to translate the kaon-physics limits into the B meson sector 10 Suppose the uncertainty in Γ(K L → ℓ + ℓ − ) could be reduced by a factor of 10, and that the central value remained unchanged. In this case, the second solution for the muon case would be strongly disfavored, given that LFUV if present at all should manifest itself as a small effect, so that χ µµ −χ ee ∼ 1.3±1.3, and, assuming MFV, C B,µµ 10 − C B,ee 10 ∼ 3.5 ± 3.5. Comparison to (23) shows that the sensitivity of thus improved K L → ℓ + ℓ − decays to C B 10 happens to be similar to the one of a tenfold reduced uncertainty of K + → π + ℓ + ℓ − to C B 9 . In either case one needs in fact more than an order-of-magnitude improvement to test the B-physics anomalies.

V. LEPTON-FLAVOR-VIOLATING DECAYS
Apart from tiny effects due to neutrino oscillations, LFV does not occur in the SM, so the decay rates can be KL → µ ± e ∓ K + → π + µ ± e ∓ KL → π 0 µ ± e ∓ K + → π + µ ± e ∓ (NA62 projection)  In the case of K + → π + µ ± e ∓ only the limit from the channel K + → π + µ + e − is considered. The last line shows the corresponding limits in the B system assuming MFV, while the rightmost column refers to the projected limit from NA62 [96].
expressed directly in terms of the NP Wilson coefficients and quark operators based on the chiral realization (13).
In general, the decay rate for K L → ℓ + 1 ℓ − 2 takes the form In the limit ℓ 1 = ℓ 2 , the vector component is absent and the expression (47) reduces to the short-distance part of (36)- (41): In the context of LFV we need ℓ 1 = µ and ℓ 2 = e where the mass of the electron has been neglected. Similarly, we find for the semileptonic decay spectra where the phase space factors are so that Based on (49) and (51), the experimental limits summarized in Table I can be turned into limits on the Wilson coefficients (|C µe 7V | 2 + |C µe 7A | 2 ) 1/2 and (|y µe 7V | 2 + |y µe 7A | 2 ) 1/2 . In particular, given that the same combination of Wilson coefficients appears if we neglect the electron mass, the analysis in terms of effective operators allows one to compare the limits from different channels in a modelindependent way (this is similar to the analysis of Higgsmediated LFV in µ → eγ and µ → e conversion in nuclei in [150]). The resulting limits are given in the first two lines of Table IV, where the limit on the C 7V,7A combination from K L → µ ± e ∓ decays is an order of magnitude more stringent than the one from K + → π + µ ± e ∓ . Even the projected improvement from NA62 [96] will fall short by a factor of 4.
As in the case of LFUV, we assume MFV to convert the limits on LFV Wilson coefficients in kaon decays to limits for the B-physics coefficients (see [151] for a similar analysis). These are shown in the bottom line of Table IV, where in the case of the K → πµe decays, the resulting constraints are slightly better than (23), but of similar order of magnitude. The strongest constraint is obtained from the limit on K L → µe.

VI. CONCLUSIONS
Motivated by the flavor anomalies observed by LHCb in semileptonic B meson decays and CMS/ATLAS in h → µτ , we presented an analysis of K → πℓ + ℓ − and K → ℓ + ℓ − decays to search for lepton flavor (universality) violation in the kaon sector. In general, the search for NP in these decays proves to be very challenging: longdistance contributions from the SM need to be separated from the interesting short-distance effects, both of which enter in poorly known low-energy constants of the χPT 3 expansion.
We observed that in the context of LFUV, this complication is absent if the difference between electron and muon parameters is considered. This simplification is due to the fact that in the SM all interactions (except those involving Higgs-Yukawa couplings) are LFU conserving. Since the Higgs corrections are negligible, it follows that the SM decays of kaons to muons or electrons differ only by phase space factors. Thus, any deviation from the SM predictions must be related to LFUV NP which is necessarily short distance once the new particles are assumed to be heavy.
For vector and axial-vector effective operators, we extracted the corresponding limits on the Wilson coefficients of the LFUV operators from K + → π + ℓ + ℓ − and K L → ℓ + ℓ − . Assuming MFV, we translated the derived limits to the corresponding B-physics Wilson coefficients. We found that the kaon limits would need to be improved by at least an order of magnitude in order to probe the parameter space relevant for the explanation of the B meson anomalies and thereby test those anomalies within the MFV hypothesis.
For the charged K-decay, improvements in this direction could be realized at the NA62 experiment, which in our view provides additional motivation to study rare decays besides the main K + → π + νν channel. Constraining LFUV in the neutral decays K L,S → π 0 ℓ + ℓ − proves to be challenging, especially since Br[K L → π 0 ℓ + ℓ − ] has not been measured and improved information from the K S → π 0 ℓ + ℓ − spectrum would be required to interpret the K L branching ratio. The alternative search channel K L → ℓ + ℓ − in principle provides access to the axialvector couplings, but also here improvements by an order of magnitude would be required. The KOTO experiment, mainly motivated by a measurement of K L → π 0 νν, might have the required sensitivity to probe LFUV in the neutral decay if the experiment could be adapted to allow for the detection of the charged leptons in the final state.
Finally, we expressed the decay rates for the LFV decay channels in terms of the corresponding Wilson coefficients and derived the bounds implied by the present experimental limits. We found that all channels are sensitive to the same combination of Wilson coefficients, with the most stringent bounds presently from K L → µ ± e ∓ .
We conclude that the upcoming NA62 experiment might have the potential to provide interesting insights into current puzzles in the flavor sector, complementary to direct measurements in B meson decays. From our analysis, the following scenarios emerge: if NP explanations for the B meson anomalies satisfied MFV, then one should see a signal at the sensitivities discussed in this paper. On the other hand, if the searches at a sensitivity expected from MFV turned out negative or if one saw a signal at current or slightly improved sensitivity, one could immediately infer that any NP scenario explaining the B anomalies would require violations of the MFV hypothesis.