Constraints on 2l2q operators from $\mu - e$ flavour-changing meson decays

We study lepton flavour violating two- and three-body decays of pseudoscalar mesons in Effective Field Theory (EFT). We give analytic formulae for the decay rates in the presence of a complete basis of QED and QCD-invariant operators. The constraints are obtained at the experimental scale, then translated to the weak scale via one-loop RGEs. The large RG-mixing between tensor and (pseudo)scalar operators weakens the constraints on scalar and pseudoscalar operators at the weak scale.


I. INTRODUCTION
The discovery of neutrino oscillations [1,2] established nonzero neutrino masses and mixing angles [3]. If neutrinos are taken massless in the Standard Model (SM), then new physics (NP) is required to explain the oscillation data. There are several possibilities to search for NP signatures, such as looking for new particles at the LHC [4,5]. Another possibility is to look for new processes among known SM particles, such as charged lepton flavor violation (CLFV) [6,7], which we define to be a contact interaction that changes the flavor of charged leptons. If neutrinos have renormalizable masses via the Higgs mechanism, then their contribution to CLFV rates is suppressed via the Glashow-Iliopoulos-Maiani mechanism by a factor ∝ ðm ν =M W Þ 4 ∼ 10 −48 . However, various extensions of the Standard Model that contain heavy new particles (see e.g., Refs. [6][7][8][9] and references therein), can predict CLFV rates comparable to the current experimental sensitivities. Indeed, low-energy precision experiments searching for forbidden SM modes, are sensitive to NP scales ≫ TeV [6]. Many experiments search for CLFV processes; for example, the μ ↔ e flavor change can be probed in the decays μ → eγ [10] and μ → 3e [11,12], in μ → e conversions on nuclei [13][14][15] and also in meson decays such as K; D; B →μe [3,[16][17][18][19][20][21][22].
In this paper, we focus on leptonic and semileptonic pseudoscalar meson decays with a μ AE e ∓ in the final state [3]. We assume that these decays could be mediated by two-lepton, two-quark contact interactions, induced by heavy new particles at the scale Λ NP > m W . The contact interactions are included in a bottom-up effective field theory (EFT) [23][24][25] approach, as a complete set of dimension-six, QED × QCD-invariant operators [6], containing a muon, an electron and one of the quark-flavorchanging combinations ds, bs, bd or cu.
Many studies on related topics can be found in the literature. The experimental sensitivity to the coefficients of four-fermion operators (sometimes referred to as oneoperator-at-a-time bounds), evaluated at the experimental scale, has been compiled by various authors [26][27][28]. Reference [29] compared the sensitivities of the LHC vs low-energy processes, to quark flavor-diagonal scalar operators. The constraints on combinations of lepton-flavorchanging operator coefficients, which can be obtained from the decays of same-flavor mesons, were studied in Ref. [30], and the radiative decays of B, D and K mesons were discussed in Ref. [31]. Lepton flavor-conserving, but quark flavor-changing meson decays (which occur in the Standard Model), are widely studied [32]. In particular, B decays attract much current interest, due to the observed anomalies [33][34][35][36][37] which suggest lepton universality violation [38][39][40][41][42][43][44]. Lepton flavor changes have been widely studied in various models (see e.g., references in Refs. [6,7,45]). More modelindependent studies, that take into account loop corrections (or equivalently, renormalization group running) have also been performed for the μ ↔ e flavor change [46,47]. Finally, with respect to the calculations in this manuscript, the leptonic branching ratio of pseudoscalar mesons is well known, and can be found in Refs. [26,28,48,49] and semileptonic branching ratios in various scenarios can be found in Refs. [50][51][52][53][54][55][56][57][58].
The aim of this paper is to obtain constraints on the operator coefficients describing meson decays at the experimental scale, and then transport the bounds to the weak scale [59]. The four-fermion operators that could induce the meson decays are listed in Sec. II. Section III gives the branching ratios for the leptonic and semileptonic decays as a function of the operator coefficients. In Sec. IV, we then use the available bounds to constrain the coefficients at the experimental scale (Λ exp ∼ 2 GeV) by computing a covariance matrix, which allows us to take into account the interferences among the operators. The bounds are then evolved from the experimental scale to the weak scale (Λ W ∼ m W ) in Sec. V, using the renormalization group equations (RGEs) of QED and QCD for four-fermion operators [46,47]. As discussed in the final section, these equations give a significant mixing of tensor operators to the (pseudo)scalars between Λ exp and Λ W , which significantly weakens the bounds on (pseudo)scalar coefficients at Λ W .

II. A BASIS OF μ − e INTERACTIONS AT LOW ENERGY
We are interested in four-fermion operators involving an electron, a muon and two quarks of different flavors, which are constructed with chiral fermions, because the lepton masses are frequently neglected, and it simplifies the matching at the weak scale onto SU(2)-invariant operators. The operators are added to the Lagrangian as where the subscript O identifies the Lorentz structure, the superscript ζ ¼ l 1 l 2 q i q j gives the flavor indices, and both run over the possibilities in the lists below, extrapolated from Refs. [6,60]:  (2) and (3) transform a muon into an electron; the e → μ operators arise in the þH:c: term of Eq. (1). So in these conventions, the lepton flavor indices are always eμ, and do not need to be given. In the following sections, we give the decay rates of pseudoscalar mesons, composed of constituent quarksq i and q j , into e þ μ − or e − μ þ . Then we obtain constraints on the operator coefficients by comparing to the experimental upper bounds on the branching ratios, e.g., BRðP 1 → e AE μ ∓ Þ ¼BRðP 1 → e þ μ − ÞþBRðP 1 → e − μ þ Þ < … which we assume apply independently to both decays. This gives independent and identical bounds on ϵ eμq i q j and ϵ eμq j q i .
In this work, we choose an operator basis with nonchiral quark currents, which is convenient for the nonchiral hadronic matrix elements involved in meson decays. Thus, the operators describing the contact interactions that can mediate leptonic (q i q j →μe) and semileptonic (q i → q jμ e) CLFV pseudoscalar meson decays at a scale Λ exp ∼ 2 GeV (Λ exp ∼ m b ≃ 4.2 GeV for bs and bd) are written as where q i;j ∈ fu; d; s; c; bg, P X ¼ P R;L ¼ 1AEγ 5 2 and σ μν ¼ i 2 ½γ μ ; γ ν . In this case, the coefficients ϵ of the operators in Eq. (4) are ϵ eμq i q j S;X In the next section, we compute the branching ratio for the (semi)leptonic decays as a function of the coefficients of Eq. (5).

III. LEPTONIC AND SEMILEPTONIC PSEUDOSCALAR MESON DECAYS
There are a multitude of bounds on rare meson decays coming from precision experiments [3,28]. The aim of this paper is to use these bounds to constrain the coefficients of Eq. (5). Thus, in this section, we compute the leptonic SACHA DAVIDSON and ALBERT SAPORTA PHYS. REV. D 99, 015032 (2019) 015032-2 and semileptonic pseudoscalar meson decay branching ratio as a function of these coefficients.

A. Leptonic decay branching ratio
We are interested in decays such as P 1 → l 1l2 where fl 1 ; l 2 g are leptons of mass m 1 , m 2 and P 1 is a pseudoscalar meson of mass M (P 1 ∈ fK 0 L ðd sþsd ffiffi 2 p Þ; D 0 ðūcÞ; B 0 ðbdÞg). In the presence of new physics, the leptonic decay branching ratio of a pseudoscalar meson P 1 of mass M is written as [26,28,49] where C 2body ¼ , m 1;2 are the masses of the leptons and τ is the lifetime of P 1 . For simplicity, we dropped the flavor superscript (ζ ¼ l 1 l 2 q i q j ) of the coefficients.
The expectation values of the quark current for a pseudoscalar meson are written as [28,49] where m i;j are the masses of the quarks, f P 1 is the decay constant of the meson and k μ is the momentum of the meson. These formulas are used for pions, kaons, and D and B mesons. The values of the constants are given in Appendix A. Note that tensor operators do not contribute to the leptonic decay, because the trace of the product of the Dirac matrices contained in the tensor operator vanishes in this case.

B. Semileptonic decay branching ratio
We are interested in decays such as P 1 → l 1l2 P 2 where fl 1 ; l 2 g are leptons of mass m 1 , m 2 and fP 1 ; P 2 g are pseudoscalar mesons of mass M; m 3 [P 1 ∈ fK þ ðusÞ; D þ ðcdÞ; B þ ðubÞ; B þ s ðsbÞg and P 2 ∈ fπ þ ðudÞ; K þ ðusÞg]. The semileptonic decay branching ratio is written as [61] where q ¼ ðp 1 þ p 2 Þ is the transferred momentum, θ is the angle between the direction of propagation of the lighter meson (P 2 ) and the antilepton (l 2 ) in the lepton's reference frame, τ and J are the lifetime and the spin of P 1 and jMj 2 is the matrix element of the semileptonic decay. The Källén function is defined as λðx; y; zÞ ¼ ðx − y − zÞ 2 − 4yz.

IV. COVARIANCE MATRIX
In this section, we use the branching ratios (BRs) of Eqs. (6) and (8) to compute a covariance matrix, that will give constraints on the coefficients that account for possible interferences. We note that BR exp 2

[BR exp
3 ] is the experimental upper limit on the leptonic decay P 1 →l 1 l 2 [semileptonic decay P 1 → P 2l1 l 2 ] branching ratio and M 2 [M 3 ] is the associated covariance matrix.
We can write the decay branching ratio of Eqs. (6) and (8) in the form where ⃗ ϵ T ð⃗ ϵÞ is a row (column) vector of coefficients, and M −1 is the inverse of the covariance matrix. The explicit forms of the 4 × 4 and 6 × 6 matrices are given in Appendix D. The diagonal elements of the covariance matrix M represents the squared bounds on our coefficients, and the off-diagonal elements represent the correlations between coefficients.

A. Bounds on the coefficients
In this section, we give constraints on the coefficients for the kaon, D and B meson leptonic and semileptonic decays. As explained in Sec. III, tensor operators do not contribute to the leptonic decays of mesons. Thus, the available upper limits on leptonic [semileptonic] pseudoscalar meson branching ratios will give constrains on the ϵ P;X and ϵ A;X [ϵ S;X , ϵ V;X and ϵ T;X ] coefficients. Indeed, hadronic matrix elements with scalar, vector or tensor quark current structure vanish in the leptonic case, while hadronic matrix elements with pseudoscalar or axial structure vanish in the semileptonic case. We consider the CLFV decays with the associated experimental upper limits given in Table I [3].

Decay
Leptonic Semileptonic The bounds in Table I will be used to constrain the coefficients at Λ exp and at Λ W after the RGE evolution of the coefficients (see Sec. V). The covariance matrices at Λ exp for the (semi)leptonic meson decays are given in Appendix E, and the bounds on coefficients are summarized in Tables II-IV.

V. RENORMALIZATION GROUP EQUATIONS
In this section, we review the evolution of operator coefficients from the experimental scale (Λ exp ∼ 2 GeV) up to the weak scale (Λ W ∼ 80 GeV) via the one-loop RGEs of QED and QCD [46,47]. We only consider the QED × QCD-invariant operators of Eq. (4). The matching onto the SMEFT basis [62] and the running above m W [63] will be studied at a later date.
A. Anomalous dimensions for meson decays Figure 1 illustrates some of the one-loop diagrams that renormalize our operators below the weak scale. Operator mixing is induced by photon loops, whereas the QCD corrections only rescale the S, P and T operator coefficients. After including one-loop corrections in the MS scheme, the operator coefficients will run with scale μ according to [46] where Γ e and Γ s are the QED and QCD anomalous dimension matrices and ⃗ ϵ is a row vector that contains the operator coefficients of Eq. (5). In this work, we use the approximate analytic solution [64] of Eq. (12) to compute the running and mixing of the coefficients between Λ exp and Λ W : where I and J represent the super-and subscripts which label operator coefficients, λ encodes the QCD corrections, andΓ e JI is the "QCD-corrected" one-loop, anomalous dimension matrix for QED [65,66]. The elements ofΓ e JI are defined as TABLE II. Constraints on the dimensionless four-fermion coefficients ϵ l 1 l 2 q i q j P;X and ϵ l 1 l 2 q i q j S;X at the experimental (Λ exp for K and D meson decays and Λ m b for B meson decays) and weak (Λ W ) scale after the RGE evolution. The last two columns are the sensitivities, or SO-at-a-time bounds; see Sec. V D. All bounds apply under permutations of the lepton and/or quark indices.  and ϵ l 1 l 2 q i q j V;X at the experimental (Λ exp for K and D meson decays and Λ m b for B meson decays) and weak (Λ W ) scale after the RGE evolution. The last two columns are the sensitivities, SO-at-a-time bounds; see Sec. V D. All bounds apply under permutations of the lepton and/or quark indices.  IV. Constraints on the dimensionless four-fermion coefficients ϵ l 1 l 2 q i q j T X at the experimental (Λ exp for K and D meson decays and Λ m b for B meson decays) and weak (Λ W ) scale after the RGE evolution. The last two columns are the sensitivities, or SO-at-a-time bounds; see Sec. V D. All bounds apply under permutations of the lepton and/or quark indices.
where the matrix elements in Γ SPT and Γ VA are defined in Sec. V.
Combining the first and second diagrams of Fig. 1 with the wave function diagrams renormalize the scalars and pseudoscalars, while the last four diagrams mix the tensors to the scalars and pseudoscalars: Similarly, the last four diagrams mix the (pseudo)scalars into the tensors. Only the wave-function diagrams renormalize the tensors, because for the first and second diagrams γ μ σγ μ ¼ 0. We obtain Finally, for the vectors and axial vectors, there is no running, but the last four diagrams contribute to the mixing of vector and axial coefficients

B. RGEs of operator coefficients
In this section we compute the evolution of the bounds from Λ exp to Λ W . In the previous section, we found a mixing between pseudoscalar and tensor coefficients, and between vector and axial coefficients. Thus, the coefficients that contributed only to the leptonic (semileptonic) decays at Λ exp will also contribute to the semileptonic (leptonic) decays at Λ W via the mixing. The matrices describing the evolution of the coefficients from Λ exp to Λ W for all the decays were obtained with Eq. (13) and are given in Appendix C.

C. Evolution of the bounds
In order to constrain the coefficients at Λ W , the constraints needs to be expressed in terms of coefficients at Λ W . However, the mixing of the pseudoscalar (axial) into the tensor (vector), and vice versa, implies that leptonic and semileptonic branching ratios can both depend on any of the ten coefficients, which we arrange in a vector as The 10 × 10 matrix we need to invert to compute the bounds at Λ W is now written as where M −1 2 and M −1 3 are the 4 × 4 and 6 × 6 matrices defined in Appendix D that we inverted to obtain the bounds at Λ exp (see Sec. IV) and R has the form of the matrices defined in Eqs. (C1), (C2) and (C3). Finally, Eq. (11) is written in the new basis as where ⃗ ϵ 0 is the vector of coefficients at Λ W , ðM 0 Þ −1 is the matrix in Eq. (20) and the superscript T means matrix transposition. All the covariance matrices at Λ W can be found in Appendix E. In Tables II-IV we summarize all the bounds on the coefficients at Λ exp and Λ W . In the leptonic decays, the evolution of the bounds on the pseudoscalar coefficients between Λ exp and Λ W is the most important effect of the RGEs as shown in the first two columns of the left panel of Table II. As can be seen in Eqs. (C1), (C2) or (C3), the running of the (pseudo)scalar coefficients is ∼1.6ð1.4Þ, which means that if we neglect the mixing of the tensor into (pseudo)scalar coefficients, the bounds on ϵ S and ϵ P will be better at Λ W for all the decays we considered. However, the large mixing of the tensor coefficients into the (pseudo)scalar ones [see Eqs. (16), (17) and (C1)-(C3)] weaken the bounds on pseudoscalar coefficients at Λ W for the kaon decay. This is due to the fact that the bounds on ϵ eμds T (see the first two columns of Table IV) are much weaker than the bounds on ϵ eμds P at Λ exp (see the first two columns of the left panel of Table II). Thus, the mixing of ϵ T into ϵ P will lead to weaker bounds on ϵ P at Λ W for the kaon decay.
For the D, B and B s meson decays, the bounds on ϵ P are a bit closer to the bound on ϵ T at Λ exp . Even with the large mixing of the tensor into the pseudoscalar coefficients, the bounds on ϵ eμcu P , ϵ eμbd P and ϵ eμbs P will be slightly better at Λ W because the running will be larger than the mixing.
In the semileptonic decays, there is also a mixing between scalar and tensor coefficients, but the bounds on scalar coefficients at Λ W increase a bit because, similarly to ϵ eμcu P , ϵ eμbd P and ϵ eμbs P , the bounds on all the scalar coefficients (first two columns of the right panel of Table II) are close to the bounds on the tensor coefficients at Λ exp . The running of the scalars will be stronger than the mixing of the tensors into the scalars, and thus, the bounds on ϵ S are better at Λ W for all the decays.
For the axial and vector coefficients, there is no running and the mixing is small. The bounds on ϵ eμds A and ϵ eμds V at Λ exp are very close (see Table III); this explains why there is no evolution of these bounds at Λ W . However, for the D, B and B s decays, the bounds on ϵ A are much weaker than the bounds on ϵ V at Λ exp , especially for the B and B s decays. Thus, the bounds on ϵ eμcu A , ϵ eμbd A and ϵ eμbs A do not evolve significantly at Λ W , but the mixing of the axial into vector coefficients will lead to weaker bounds on ϵ eμcu V , ϵ eμbd V and ϵ eμbs V at Λ W as shown in the first two columns of the two panels of Table III.
Finally, the running of tensor coefficients is tiny, and the mixing of the (pseudo)scalar coefficients into the tensor ones is small. Thus, the evolution of the bounds is small for the tensor coefficients (first two columns of Table IV) similarly to the bounds on vector and axial coefficients in the kaon decay (first two columns of Table III). Finally, the matching at Λ W along with the evolution of the bounds between Λ W and Λ NP will be given in a future publication [67].

D. Single operator approximation
We also computed the sensitivities of the various decays to the coefficients at Λ exp , and these are given in the third columns of Tables II to IV. The sensitivity is the value of the coefficient below which it could not have been observed, and is calculated as a "single operator" (SO)-at-a-time bound, that is by allowing only one nonzero coefficient at a time in the branching ratio [see Eqs. (6) and (9)]. This is different from setting bounds on coefficients (first two columns of Tables II to IV), which are obtained with all coefficients nonzero, and exclude the parameter space outside the allowed range. It is clear that the sensitivities are sometimes an excellent approximation to the bounds, and sometimes differ by orders of magnitude.
To compute the evolution of the sensitivities of the decays to the coefficients at Λ W (given in the last columns of Tables II-IV) inverting the product and taking the square root will give the sensitivity of the decay to the coefficient at Λ W .

E. Updating the bounds
In future years, the experimental data on LFV meson decays could improve, so in this section, we consider how to update our bounds, without inverting large matrices.
The bounds on coefficients at Λ exp obtained in this work are of the form jϵj < ffiffiffiffiffiffiffiffiffiffiffi ffi BR exp p × constant. Thus, all the bounds at Λ exp given in Tables II-IV can be updated by  rescaling by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðBR exp new Þ=ðBR exp old Þ p when the data improves. However, in principle, the 10 × 10 matrix of Eq. (20) must then be inverted to obtain the bounds at Λ W . So we now describe approximations that allow to obtain the bounds at Λ W with manageable matrices.
For the semileptonic decay, the bounds at Λ exp can be obtained by neglecting all the interference terms between the scalar, vector and tensor coefficients of either chirality [see Eq. (9)]. The 6 × 6 matrix in Eq. (D2) then becomes diagonal and easy to invert. This approximation will give bounds at Λ exp on ϵ S;X , ϵ V;X and ϵ T;X close to those obtained in the first columns of Tables II-IV (which include the interference terms).
In the leptonic decay [Eq. (6)], a reasonable approximation for the bounds at Λ exp is to keep the interference between axial and pseudoscalar coefficients of opposite chirality [with m 2 ¼ m μ in Eq. (6)]. The other interference terms, proportional to m 1 ¼ m e , can be neglected. Thus, bounds on ϵ A and ϵ P at Λ exp , which are a reasonable approximation to the first columns of Tables II and III, can be obtained by inverting a 2 × 2 matrix in the basis ðϵ P;X ; ϵ A;Y Þ where X ∈ L, R and Y ∈ R, L, instead of the 4 × 4 matrix in Eq. (D1).
To obtain bounds at Λ W , it is necessary to keep the mixing between ϵ S , ϵ P , ϵ T , and between ϵ V and ϵ A . Then, the bounds on ϵ S , ϵ P , ϵ T , ϵ V and ϵ A at Λ W can be obtained by considering M −1 0 in Eq. (20) as a product of 5 × 5 matrices in the basis (ϵ P;X , ϵ S;X , ϵ T;X ,ϵ V;Y ,ϵ A;Y ) where X and Y are the chiralities. However, ϵ S , ϵ P and ϵ T must have the same chirality, but different from the chirality of ϵ V and ϵ A in order to take into account the mixing induced by the RGEs, that occurs only for coefficients of the same chirality [see Eqs. (13) and (C1)-(C3)]. This is due to the fact that it is necessary to keep the interference between axial and pseudoscalar coefficients of different chiralities to compute the bounds on ϵ P;X and ϵ A;Y .

VI. CONCLUSION
In this paper, we considered operators which simultaneously change lepton and quark flavor, and obtain constraints on the coefficients using available data on (semi)leptonic pseudoscalar meson decays. Section II listed the dimension-six, two lepton-two quark operators and their associated coefficients at the experimental scale Λ exp . Scalar, pseudoscalar, vector, axial and tensor operators were included. The leptonic and semileptonic branching ratios of pseudoscalar mesons, as a function of the operator coefficients, were given in Sec. III. We found that tensor operators do not contribute to the leptonic decays but only to the semileptonic decays, in which the interference between ϵ S;L (ϵ S;R ) and ϵ T R (ϵ T L ) vanishes. The constraints on operator coefficients, evaluated at the experimental scale, are given in Tables II-IV and discussed in Sec. IV. The bounds are obtained via the appropriate covariance matrices, which allows to take into account the interferences among operators [see Eqs. (6), (9), (D1) and (D2)]. The matrices are given in Appendix B. Section V gave the renormalization group evolution of the coefficients from the experimental to the weak scale Λ W , and the formalism used to compute the covariances matrices at Λ W . The weak-scale constraints on the coefficients are given in Tables II-IV. The large mixing of tensor coefficients into (pseudo)scalar coefficients has important consequences on the evolution of the bounds on scalar and pseudoscalar coefficients. Indeed, in the case of the kaon decay, the experimental-scale bounds on tensor coefficients are significantly weaker than those on pseudoscalars. As a result, the pseudoscalar bounds are weaker at Λ W , compared to the bounds at Λ exp . The bounds on scalar coefficients at Λ W are slightly stronger than at Λ exp . There is no running for the vector and axial coefficients, due to the fact that we considered quarkflavor-changing operators, and the mixing is small, but the bounds on axial coefficients are much weaker than the bounds on vector coefficients for the D, B and B s decays. This leads to much weaker bounds on vector coefficients at Λ W . Similarly, the running and mixing of tensor coefficients are small. As a result, the bounds on the axial and tensor coefficients do not evolve significantly between the experimental and weak scales.
We conclude by noting the importance of including interferences among operators in computing the bounds on their coefficients. As shown in Sec. V D, the sensitivities of the decays to ϵ P and ϵ A obtained at Λ exp and to ϵ P , ϵ A and ϵ V at Λ W in the single-operator approximation are better by several orders of magnitude compared to the bounds obtained by keeping the interferences among operators. We found that the renormalization group running between the experimental and weak scales has an important effect on the evolution of the bounds, especially the large mixing of the tensor (axial) into the pseudoscalar (vector), which lead to weaker bounds on pseudoscalar (vector) coefficients at Λ W for the kaon (D, B and B s ) decay.

APPENDIX A: CONSTANTS
In this appendix, we give all the constants used in our calculations: SACHA DAVIDSON and ALBERT SAPORTA PHYS. REV. D 99, 015032 (2019) 015032-8 All the masses and lifetimes can be found in Ref. [3].

APPENDIX B: KINEMATICS AND FORM FACTORS FOR SEMILEPTONIC DECAYS
In this appendix, we give the form factor and the detailed scalar product of Eq. (9). The q 2 dependence of the form factors for the kaon is given by [50] f Kπ and for the D and B mesons they are given by [51,52] where λ þ;0 are constants, and m J P is the mass of the lightest resonance with the right quantum numbers to mediate the transition (D þ s and D Ãþ s for example). We took q 2 ¼ q 2 max ¼ ðM − m 3 Þ 2 to compute the form factors f þ , f − and f 0 . All these values can be found in Appendix A.

APPENDIX C: RGEs
In this appendix, we give the 10 × 10 matrices obtained with Eq. (13) we used to obtain the bounds at Λ W [with Eq. (20)]. For the decay of light quarks (kaon and D meson decays), the experimental scale is taken as 2 GeV because most of the time, it is the renormalization scale chosen to obtain the lattice form factors.
The evolution of the coefficients (ϵ eμds ) involved in the kaon decays is given by For the D meson decays, the evolution of the coefficients (ϵ eμcu ) is given by  In the B and B s meson decays, the reference scale is the b quark mass (Λ m b ∼ 4.18 GeV). Thus, the evolution of the coefficients (ϵ eμbd and ϵ eμbs ) is slightly smaller. In fact, in Eq. (13), the part with the anomalous dimension that gives the matrix element in Eq. (C1) is multiplied by a factor logð Λ W Λ m b Þ=logð Λ W Λ exp Þ ∼ 0.8. Moreover, the strong coupling constant at Λ m b will also be smaller [α s ðΛ m b Þ ∼ 0.23 and α s ðΛ exp Þ ∼ 0.3]. Thus, for the B and B s meson decays, the evolution of the coefficients (ϵ eμbd and ϵ eμbs ) is given by 0

APPENDIX D: COVARIANCE MATRIX
In this appendix, we give details of the formalism introduced in Eq. (11) of Sec. IV. The matrices in the basis ðϵ P;L ; ϵ A;L ; ϵ P;R ; ϵ A;R Þ and ðϵ S;L ; ϵ V;L ; ϵ T L ; ϵ S;R ; ϵ V;R ; ϵ T R Þ are written as Inverting M −1 2 (M −1 3 ) will give the bounds on the coefficients involved in the leptonic (semileptonic) decays. Finally, note that for semileptonic kaon and D meson decays, the experimental upper limits are not the same for μ þ e − and μ − e þ in the final state. In this case, we sum the M −1 3 for each bound and then invert it to obtain the covariance matrix of Sec. IV. The matrix elements of Eq. (D1) are written as For simplicity we note that dϕ ¼ and the matrix elements of Eq. (D2) are written as APPENDIX E: COVARIANCE MATRICES AT Λ exp AND Λ W In this appendix, we give the covariance matrices at Λ exp and Λ W , after the RGE evolution.

Kaon decays
Using the upper limit of Table I, for the leptonic kaon decays, we compute the associated covariance matrix in the basis ðϵ eμds P;L ; ϵ eμds A;L ; ϵ eμds P;R ϵ eμds A;R Þ: Then we use the bounds on the semileptonic kaon decay to compute the covariance matrix for the semileptonic decays in the basis ðϵ eμds S;L ; ϵ eμds V;L ; ϵ eμds T L ; ϵ eμds S;R ; ϵ eμds V;R ; ϵ eμds T R Þ: The diagonal elements give the bounds on jϵj 2 . The bounds on the coefficients are the square roots of the diagonal elements. For instance, ϵ eμds S;L is excluded above ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1.09 × 10 −12 p . The covariance matrix in the basis ð ϵ eμds P;L ; ϵ eμds A;L ; ϵ eμds P;R ; ϵ eμds A;R ; ϵ eμds S;L ; ϵ eμds V;L ; ϵ eμds T L ; ϵ eμds S;R ; ϵ eμds V;R ; ϵ eμds