Constraints on 2l2q operators from $μ- e$ flavour-changing meson decays
Sacha Davidson Albert Saporta










             



Addressable electron spin resonance using donors and donor molecules in silicon
    Michelle Y Simmons
    
===============================================================================




  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×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.






§ INTRODUCTION



      The discovery of neutrino oscillations <cit.>

established non zero neutrino masses  and  mixing angles <cit.>.
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 <cit.>. Another possibility
is to look  for new processes among
known SM particles, such as Charged Lepton Flavour Violation
(CLFV) <cit.>, which we define  to be a contact
interaction that changes the flavour of charged leptons.
If neutrinos have renormalizable masses via  the Higgs mechanism,
then their contribution to CLFV rates is GIM  suppressed by a factor ∝ (m_ν/M_W)^4 ∼ 10^-48.  
However, various  extensions of the
Standard Model that contain heavy new particles
(see e.g. <cit.>
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 <cit.>. Many
experiments search for  CLFV processes;  for example,
μ↔ e  flavour change can be probed in the decays μ→ e γ <cit.> and μ→ 3e <cit.>, in μ→ e conversion  on nuclei <cit.> and also in meson decays such as K,D,B →μ e <cit.>. 
 In this paper, we focus on  leptonic and semileptonic pseudoscalar meson decays with a μ^± e^∓ in the final state <cit.>.
    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) <cit.> approach, 
 as a complete set of dimension six,  QED×QCD-invariant operators <cit.>,
 containing   a muon, an electron and one of the  quark-flavour-changing 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 refered to as one-operator-at-a-time bounds), evaluated at the experimental scale, has been compiled by various authors <cit.>.
  Reference <cit.>   compared the sensitivities of the LHC vs
     low-energy processes, to quark flavour-diagonal scalar operators.
    The constraints  on combinations of   lepton-flavour-changing operator coefficients,
 which can be obtained from   the decays of same-flavour mesons,
 were studied in <cit.>, and  the  radiative
 decays of B,D and K mesons were discussed in <cit.>. 
        Lepton flavour-conserving, but quark flavour-changing  meson  decays (which occur in the Standard Model), are widely studied <cit.>.
    In particular,  B decays attract much current interest, due to the  observed
    anomalies
    <cit.>  which 
    suggest lepton universality violation <cit.>.
 Lepton flavour change has been widely studied in various
  models (see e.g. references of <cit.>).     
  More model-independent studies, that take into account loop corrections
  (or equivalently, renormalization group running) have also been performed
 for μ↔ e flavour change  <cit.>.
Finally, with respect to the calculations in
this manuscript, the leptonic  branching ratio of
pseudoscalar mesons  is well-known,
and  can be found in <cit.> and 
semi-leptonic branching ratios in various scenarios  can be found in <cit.>.

    
    
   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[In a future publication,  
   we will give the evolution from the weak scale  to the NP scale, and discuss the prospects for
   reconstructing   the fundamental Lagrangian of the New Physics from
   the operator coefficients.].
 The four fermion operators  that could induce the meson decays
 are listed in section <ref>.  Section <ref> 
 gives  the branching ratios for the leptonic and semileptonic  decays as a function of the operator coefficients. In section <ref>, we then use the available bounds to constrain  the coefficients at the experimental scale (Λ_exp∼2 GeV)
 by computing a covariance matrix, which allow 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 section <ref>, using
 the Renormalization Group Equations (RGEs)   of QED and QCD for
 four-fermion operators  <cit.>.
As discussed in the final section, these equations give a significant mixing of tensor operators into the (pseudo)scalars between Λ_exp and Λ_W, which significantly weakens the bounds on (pseudo)scalar coefficients at Λ_W. 






§ A BASIS OF   Μ-E INTERACTIONS  AT LOW ENERGY





The four fermion-operators are labeled with the lepton and quark flavour in superscripts and the nature of Lorentz contraction (scalar, vector, tensor) in subscripts. We also choose to have the flavour change inside a spinor contraction.



We are interested in four-fermion operators
involving an electron, a muon
and two  quark of different  flavours,
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

    δ L = + 2√(2) G_F ∑_O∑_ζ
    C _O^ζ𝒪_O^ζ  + h.c.


where  the subscript O identifies the Lorentz structure,
the superscript ζ=l_1 l_2 q_i q_j
gives the flavour indices, and both run over the
possibilities in the lists below, extrapolated from
  <cit.>: 
  
  
    [                                               𝒪^eμ uc_V,YY =(eγ^α P_Y μ ) 
    (uγ_α P_Y c ),  𝒪^eμ uc_V,YX = (eγ^α P_Y μ ) 
    (uγ_α P_X c );                                              𝒪^eμ cu_V,YY = (eγ^α P_Y μ ) 
    (cγ_α P_Y u ),   𝒪^eμ cu_V,YX = (eγ^α P_Y μ ) 
    (cγ_α P_X e);                                                   𝒪^eμ uc_S,YY = (e  P_Y μ ) 
    (u P_Y c ),       𝒪^eμ uc_S,YX = (e  P_Y μ ) 
    (u P_X c );                                                  𝒪^eμ cu_S,YY = (e  P_Y μ ) 
    (c P_Y u ) ,       𝒪^eμ cu_S,YX = (e  P_Y μ ) 
    (c P_X u );                                                   𝒪^eμ uc_T,YY = (eσ P_Y μ ) 
    (uσ P_Y c )                                             ;                                                   𝒪^eμ cu_T,YY = (eσ P_Y μ ) 
    (cσ P_Y u )                                              ]



    [                                              𝒪^eμ ds_V,YY = (eγ^α P_Y μ ) 
    (dγ_α P_Y s ),  𝒪^eμ ds_V,YX = (eγ^α P_Y μ ) 
    (dγ_α P_X s );                                              𝒪^eμ sd_V,YY = (eγ^α P_Y μ ) 
    (sγ_α P_Y d ),  𝒪^eμ sd_V,YX = (eγ^α P_Y μ ) 
    (sγ_α P_X d );                                                   𝒪^eμ ds_S,YY = (e  P_Y μ ) 
    (d P_Y s ),       𝒪^eμ ds_S,YX = (e  P_Y μ ) 
    (d P_X s );                                                   𝒪^eμ sd_S,YY = (e  P_Y μ ) 
    (s P_Y d ),       𝒪^eμ ds_S,YX = (e  P_Y μ ) 
    (d P_X s );                                                   𝒪^eμ ds_T,YY = (eσ P_Y μ ) 
    (dσ P_Y s )                                             ;                                                   𝒪^eμ sd_T,YY = (eσ P_Y μ ) 
    (sσ P_Y d )                                             ;                                              ]

 
where
YY ∈{LL,RR}, and  X Y  ∈{LR,RL}, and
the list is given explicitly for the Kaon and D-meson operators.
The lists for   the B_d  and B_s are
obtained from eqn. (<ref>) by replacing
ds → db,sb.
The operators
  are normalised such that the Feynman
rule will be +i C/Λ^2. 
The operators in the lists (<ref>) and (<ref>) 
transform a muon into an electron; the e →μ
operators arise  in the +h.c. of eqn (<ref>). So in
these conventions, the lepton flavour 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 quarks
q̅_i and q_j, into  e^+ μ^- or e^- μ^+. Then
we obtain constraints on the operator coefficients by
comparing to the experimental upper bounds on
the branching ratios, e.g.
BR(P_1 → e^±μ^∓) =
BR(P_1 → e^+ μ^-) +
BR(P_1 → e^- μ^+) < ...
which we suppose to apply independently to
both decays. This gives  independent and identical bounds
on ϵ^eμ q_iq_j and  ϵ^eμ q_jq_i. 



In this work, we choose an operator basis  with non-chiral quark currents, which is convenient for the non-chiral hadronic matrix elements involved in meson decays.  Thus, the operators describing the contact interactions that can mediate leptonic (q_iq_j →μe) and semileptonic (q_i → q_jμe) CLFV pseudoscalar meson decays  at a scale Λ_exp∼ 2 GeV  (Λ_exp∼ m_b ≃  4.2 GeV for bs and bd) are written: 
 
    𝒪^ e μ q_i q_j _S,X = ( e  P_Xμ) 
    ( q_i    q_j ),   𝒪^eμ q_i q_j _P,X = ( eP_Xμ) 
    ( q_i γ^5   q_j ) 
       𝒪^eμ q_i q_j _V,X = ( eγ^α P_Xμ) 
    ( q_iγ_α q_j ),   𝒪^eμ q_i q_j _A,X = ( eγ^α P_Xμ) 
    ( q_iγ_αγ^5  q_j )  
       𝒪^e μ q_i q_j _T_X = ( eσ^αβ P_Xμ) ( q_i σ_αβP_X  q_j )
 


where  q_i,j∈{u,d,s,c,b}, P_X= P_R,L = 1 ±γ_5/2 and σ^μν = i/2[γ^μ,γ^ν]. 








In this case, the coefficients ϵ of the operators  in eqn.  (<ref>)   are 
 : 



    ϵ_S,X^eμ q_i q_j  =1/2(C_S,XR^eμ  q_i q_j +C_S,XL^eμ  q_i q_j ) ,   ϵ_P,X^eμ  q_i q_j  =1/2(C_S,XR^eμ q_i q_j -C_S,XL^eμ  q_i q_j ) 
       ϵ_V,X^eμ  q_i q_j  =1/2(C_V,XR^eμ  q_i q_j +C_V,XL^eμ  q_i q_j ) ,   ϵ_A,X^eμ  q_i q_j  =1/2(C_V,XR^eμ  q_i q_j -C_V,XL^eμ  q_i q_j ) 
       ϵ_T,X^eμ  q_i q_j  =C_T,XX^eμ  q_i q_j









In the next section, we compute the branching ratio for the (semi)leptonic decays as a function of  the coefficients of eqn. (<ref>). 



    C_S,X^l_1 l_2 q_i q_j /Λ_NP^2 =1/2(C_S,XR^l_1 l_2 q_i q_j +C_S,XL^l_1 l_2 q_i q_j ) = 4 G_F/√(2)ϵ_S,X^l_1 l_2 q_i q_j ,   C_P,X^l_1 l_2 q_i q_j /Λ_NP^2=1/2(C_S,XR^l_1 l_2 q_i q_j -C_S,XL^l_1 l_2 q_i q_j ) = 4 G_F/√(2)ϵ_P,X^l_1 l_2 q_i q_j 
       C_V,X^l_1 l_2 q_i q_j /Λ_NP^2=1/2(C_V,XR^l_1 l_2 q_i q_j +C_V,XL^l_1 l_2 q_i q_j ) = 4 G_F/√(2)ϵ_V,X^l_1 l_2 q_i q_j ,   C_A,X^l_1 l_2 q_i q_j /Λ_NP^2=1/2(C_V,XR^l_1 l_2 q_i q_j -C_V,XL^l_1 l_2 q_i q_j ) = 4 G_F/√(2)ϵ_A,X^l_1 l_2 q_i q_j 
       C_T_X^l_1 l_2 q_i q_j /Λ_NP^2 = 4 G_F/√(2)ϵ_T_X^l_1 l_2 q_i q_j



The lagrangian of the bottom-up EFT at Λ_exp is given by  : 


    ℒ_eff = ℒ_SM + ∑_OC_O^l_1 l_2 q_i q_j 𝒪^l_1 l_2 q_i q_j _O + h.c.


where O ∈{S,P,V,A,T}. 


To obtain bounds on these coefficients, we need the available data  that can probe the four fermion operators of eqn. (<ref>). We thus need to compute an observable that can be constrained experimentally as a function of these coefficients.
Notice that  experimental upper limits on branching ratios are often given in the form BR(P_1 → (P_2) l_1^±l_2^∓)=BR(P_1 → (P_2) l_1^+l_2^-)+BR(P_1 → (P_2) l_1^-l_2^+) where P_1 and P_2 are pseudoscalar meson. We thus suppose that these bounds to apply
independently to both decays.
This gives independent and identical bounds
on ϵ^l_1 l_2 q_i q_j and ϵ^l_2 l_1 q_i q_j. 




§ LEPTONIC AND SEMILEPTONIC PSEUDOSCALAR MESON DECAYS



There are a multitude of bounds   on rare meson decays coming from precision experiments <cit.>. The aim of this paper is to use these bounds to constrain the coefficients of eqn. (<ref>). Thus, in this section, we compute the leptonic and semileptonic pseudoscalar meson decay branching ratio as a function of these coefficients.



 §.§ Leptonic decay branching ratio

We are interested in decays such as :
P_1 → l_1 l̅_2  where {l_1, l_2 }  are leptons of mass m_1,m_2 and P_1 is a pseudoscalar meson of mass M (P_1 ∈{K^0_L(d̅s+s̅d/√(2)),D^0(u̅c),B^0(b̅d) }).  
In the presence of New Physics, the leptonic decay branching ratio of a pseudoscalar meson P_1 of mass M is written <cit.>:

 
    BR(P_1 → l_1 l̅_2  )/C_2body   =
    (|ϵ_P,L|^2+|ϵ_P,R|^2) P̃'̃^2 (M^2 -m_1^2-m_2^2) 
       + (|ϵ_A,L|^2+|ϵ_A,R|^2) Ã'̃^2[(M^2-m_1^2-m_2^2)(m_1^2+m_2^2) +4m_1^2m_2^2] 
       - 2(ϵ_P,Lϵ_A,R + ϵ_P,Rϵ_A,L)P̃'̃Ã'̃
    m_2(M^2 +m_1^2 -m_2^2)  
       + 2(ϵ_P,Lϵ_A,L + ϵ_P,Rϵ_A,R)P̃'̃Ã'̃
    m_1(M^2 +m_2^2 -m_1^2) 
       -4ϵ_P,Lϵ_P,RP̃'̃^2
    m_1m_2 
       -4ϵ_A,Lϵ_A,RÃ'̃^2
    M^2m_1m_2
 




where C_2body =τ r^*G_F^2/π M^2, r^*= 1/2M√((M^2-(m_1+m_2)^2)(M^2-(m_1-m_2)^2)), m_1,2 are the masses of the leptons and τ is the lifetime of P_1. For simplicity, we dropped the flavour superscript (ζ=l_1 l_2 q_i q_j) of the coefficients. 


The expectation values of the quark current for a pseudoscalar meson  are written <cit.> : 


    P̃'̃=1/2⟨0|q̅_̅i̅γ^5 q_j|P_1|=⟩f_P_1 M^2/2(m_i +m_j),   Ã'̃k̃^̃μ̃=1/2⟨0|q̅_̅i̅γ^μγ^5 q_j|P_1|=⟩f_P_1k^μ/2


where m_i,j are the masses of the quarks, f_P_1  is the decay constant of the meson and k^μ  the momentum of the meson. 
These formulae are used for pions, Kaons, D and B mesons. The values of the constants are given in appendix <ref>. Note that tensor operators do not contribute to the leptonic decay, because the trace of product of the Dirac matrices contained in the tensor operator vanishes in this case.




 §.§ Semileptonic decay branching ratio




We are interested in decays such as :
P_1 → l_1 l̅_2   P_2  where {l_1, l_2 }  are leptons of mass m_1,m_2 and {P_1,P_2} are pseudoscalar mesons of mass M,m_3 (P_1 ∈{K^+(us̅),D^+(cd̅),B^+(ub̅),B_s^+(sb̅) } and P_2 ∈{π^+(ud̅),K^+(us̅) }). The semileptonic decay branching ratio is written <cit.>:

    BR(P_1 → l_1 l̅_2   P_2)=τ/512π^3M^31/2J+1∫_(m_1+m_2)^2^(M-m_3)^2dq^2 ∫_-1^1dcosθ|ℳ|^2√(λ(M^2,m_3^2,q^2))√(λ(q^2,m_1^2,m_2^2))/q^2

 

where q=(p_1 + p_2) is the transferred momentum, θ the angle between the direction of propagation of the lighter meson (P_2) and the antilepton (l_2) in the leptons reference frame, τ and J  the lifetime and the spin  of P_1  and |ℳ|^2 the matrix element of the semileptonic decay. The Kallen function is defined as λ(x,y,z) = (x-y-z)^2 - 4yz.





In the presence of New Physics, the matrix element in the semileptonic decay branching ratio of eqn. (<ref>) is written : 



    |ℳ|^2/8 G_F^2   = 2(|ϵ_S,L|^2+|ϵ_S,R|^2) S̃^2 (p_1.p_2) 
        + 1/4(|ϵ_V,L|^2 + |ϵ_V,R|^2 ) [f_+^2 (
     4(p_1.P)(p_2.P) - 2P^2 (p_1.p_2) ) + f_-^2 (
     4(p_1.q)(p_2.q) - 2q^2 (p_1.p_2) )
        + 4f_+f_-   ((p_1.q)(p_2.P) + (p_1.P)(p_2.q)-(p_1.p_2)(P.q)    )] 
    
        + 4(|ϵ_T_R|^2 + |ϵ_T_L|^2 ) T̃'̃^2 [ 4 (p_1.q)(p_2.P)(P.q) + 4(p_1.P)(p_2.q)(P.q) -2(p_1.p_2)(P.q)^2  
       +2P^2q^2(p_1.p_2) - 4 P^2(p_1.q)(p_2.q) - 4q^2(p_1.P)(p_2.P) 
     ] 
       -2(ϵ_S,Lϵ_V,R+ϵ_S,Rϵ_V,L)S̃m_2[ (f_+ (p_1.P) + f_-(p_1.q) ) ] 
       +2(ϵ_S,Lϵ_V,L+ϵ_S,Rϵ_V,R)S̃m_1[ (f_+ (p_2.P) + f_-(p_2.q) ) ] 
        +8(ϵ_S,Rϵ_T_R+ϵ_S,Lϵ_T_L)S̃T̃'̃[((p_1.P)(p_2.q)-(p_1.q)(p_2.P)  ) ] 
       -4ϵ_S,Lϵ_S,RS̃^2 m_1 m_2 
       -ϵ_V,Lϵ_V,R m_1 m_2[f_-^2q^2 + f_+^2P^2 + 2 f_+ f_-(P.q)]
       +4(ϵ_V,Lϵ_T_R +ϵ_V,Rϵ_T_L)
         T̃'̃m_2[f_+((p_1.q)p^2-(P.p_1)(P.q)) +  f_-((p_1.q)(P.q)-(p_1.P)q^2)]
        +4 (ϵ_V,Rϵ_T_R+ϵ_V,Lϵ_T_L)T̃'̃m_1[ (f_+((P^2)(p_2.q)-(p_2.P)(P.q)) +f_-((p_2.q)(P.q)-q^2(p_2.P))
     )] 
       +16ϵ_T_Rϵ_T_LT̃'̃^2 m_1 m_2[(P.q)^2-P^2q^2]
 



where  p_1,p_2,k,p_3  are respectively  the 4-momentum of the leptons 1 and 2,  and  the 4-momenta of P_1 and P_2,    P = k +p_3 and  the hadronic matrix elements are  written <cit.> : 

    Ṽ^μ =1/2⟨P_2|q̅_̅i̅γ^μq_j|P_1|=⟩1/2(P^μf_+^P_1 P_2(q^2) + q^μf_-^P_1 P_2(q^2) ) 
       S̃ =1/2⟨P_2|q̅_̅i̅   q_j|P_1|=⟩1/2(M^2-m_3^2)/(m_q_i-m_q_j)f_0^P_1 P_2(q^2)
       T̃^μν = 1/2⟨P_2|q̅_̅i̅σ^μνq_j|P_1|=⟩-i/2(f_+^P_1 P_2(q^2)-f_-^P_1 P_2(q^2))/M^*(P^μq^ν-P^νq^μ) 
       T̃'̃ = 1/2(f_+^P_1 P_2(q^2)-f_-^P_1 P_2(q^2))/M^*
 
For simplicity, we suppressed the q^2 dependence of the form factors f_+,-,0 in eqn. (<ref>),  and   the
 flavour  superscript (ζ=l_1 l_2 q_i q_j) of the coefficients.
  Notice there is no interference between ϵ_S,L (ϵ_S,R ) and ϵ_T_R (ϵ_T_L) because the  trace of the product of Dirac matrices involved in tensor and scalar operators of different chirality vanishes.  The form factors and the scalar product in eqn. (<ref>) are given in appendix <ref>. 

For simplicity, we do not give the analytic expression of the integrated semileptonic decay branching ratio, but only perform the integrals numerically. 






§ COVARIANCE MATRIX



In this section, we use  the Branching Ratios (BRs) of
  eqns (<ref>) and (<ref>) to compute a covariance matrix, that will give constraints on  the coefficients that account
   for possible interferences. We note BR_2^exp [BR_3^exp] the experimental upper limit on the leptonic decay P_1 →l̅_1l_2 [semileptonic decay P_1 → P_2l̅_1l_2] branching ratio and M_2 [M_3] the associated covariance matrix. 



We can write the decay branching ratio of eqn. (<ref>)  and (<ref>) in the form 

    ϵ⃗^TM^-1ϵ⃗ =1



where  ϵ⃗^T (ϵ⃗) is a row (column)
  vector of coefficients, and M^-1 is the inverse of the covariance matrix. The explicit form of the 4 × 4 and 6 × 6 matrices is given in appendix <ref>. The diagonal elements  of the covariance matrix M
represent the squared bounds on our coefficients, and the off-diagonals elements represent the correlation between coefficients. 












 §.§ Bounds on the coefficients


In this section, we give constraints on the coefficients for Kaon, D and B meson leptonic and semileptonic decays. As explained in section <ref>, 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 struture vanish in the semileptonic case. We consider the  CLFV decays with the associated experimental upper limits given in table <ref>
<cit.>.












    BR_2^exp(K^0_L →μ^±e^∓)<4.7 × 10^-12  <cit.>


    BR_3^exp(K^+ →π^+ μ̅e) < 1.3 × 10^-11,   
    BR_3^exp(K^+ →π^+ e̅μ) < 5.2 × 10^-10  <cit.>












The bounds in table <ref> will be used to constrain the coefficients at Λ_exp and the at Λ_W after the RGEs evolution of the coefficients (see section <ref>). The covariance matrices at Λ_exp for the (semi)leptonic meson decays are given in appendix <ref>, and the bounds on coefficients are summarized in table <ref>, <ref> and <ref>. 







    BR_2^exp(D^0 →μ^± e^∓)<1.3 × 10^-8  <cit.>


    BR_3^exp(D^+ →π^+ μ̅e) < 3.6 × 10^-6,   
    BR_3^exp(D^+ →π^+ e̅μ) < 2.9 × 10^-6  <cit.>


    BR_3^exp(D^+_S → K^+ μ̅e) < 9.7 × 10^-6,   
    BR_3^exp(D^+_S → K^+ e̅μ) < 1.4 × 10^-5  <cit.>















    BR_2^exp(B^0 →μ^±e^∓)<2.8 × 10^-9  <cit.>


    BR_3^exp(B^+ →π^+ μ^±e^∓) < 1.7 × 10^-7  <cit.>














    BR_2^exp(B^0_S →μ^±e^∓)<1.1 × 10^-8  <cit.>


    BR_3^exp(B^+ → K^+ μ^±e^∓) < 9.1 × 10^-8  <cit.>


















§ RENORMALIZATION GROUP EQUATIONS (RGES)



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 <cit.>. We only consider the QED×QCD invariant operators of eqn. (<ref>). The matching onto the SMEFT basis <cit.> and the running  above m_W <cit.> will be studied  at a later date.




 §.§ Anomalous dimensions for meson decays



Figure <ref> 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 <cit.> :


    μ∂/∂μϵ⃗= α_e/4πϵ⃗ Γ^e +α_s/4πϵ⃗ Γ^s


where  Γ^e and Γ^s are the QED and QCD anomalous dimension matrices  and ϵ⃗ is a row vector that contains  the operator coefficients  of eqn. (<ref>).  In this work, we use the approximate analytic solution   <cit.> of eqn. (<ref>) to compute the running and mixing of the coefficients between Λ_exp and Λ_W :




    ϵ_I (Λ_exp)  = ϵ_J(Λ_W)λ^a_J( δ_JI - α_eΓ^e_JI/4πlogΛ_W/Λ_exp)
 


where  I,J represent
the super- and sub-scripts which label operator
coefficients,  λ encodes the
QCD corrections, and Γ^e_JI  is the
“QCD-corrected”   one-loop, anomalous dimension matrix for QED <cit.> . 
The elements of Γ^e_JI are defined as:

    Γ^e_JI=Γ^e_JI f_JI,     f_JI=1/1+a_J - a_Iλ^a_I - a_J - λ/1 - λ
    ,   Γ^e=
    [ Γ_SPT     0;     0  Γ_VA ]  .
 

where there is no sum on I,J, 
λ = α_s(Λ_W)/α_s(Λ_exp),
 and a_J = Γ_JJ^s/2 β_0 = { -12/23, -12/23, 4/23} for J ∈{S,P,T}. 
The QED anomalous dimensions are

    Γ_SPT=
    [
    [                   γ_PP^l_1l_2q_iq_j                 0 γ_PT^l_1l_2q_iq_j;                                   0 γ_SS^l_1l_2q_iq_j γ_ST^l_1l_2q_iq_j;                   γ_TP^l_1l_2q_iq_j γ_TS^l_1l_2q_iq_j γ_TT^l_1l_2q_iq_j;                   ]]
    ,   Γ_VA=
    [
    [                   γ_AA^l_1l_2q_iq_j γ_AV^l_1l_2q_iq_j;                   γ_VA^l_1l_2q_iq_j γ_VV^l_1l_2q_iq_j;                   ]]


where the matrix elements in Γ_SPT

 and Γ_VA 
are defined in section <ref>. 






















For the scalars and pseudoscalars,
the  wavefunction, first and   second  diagrams  of figure <ref>
renormalize the  coefficients,
while the last four diagrams mix the tensors  into the scalars  and pseudoscalars:   



    γ_ SS^q, q =
      [                     ϵ_ S, L^q q     ϵ_ S, R^q q;     ϵ_ S, L^q q - 6 (1 + Q_q^2)               0;     ϵ_ S, R^q q               0 - 6 (1 + Q_q^2);                 ]   γ_ TS^q, q =
      [             ϵ_ S, L^q q ϵ_ S, R^q q; ϵ_ T, L^q q      48 Q_q           0; ϵ_ T, R^q q           0      48 Q_q;             ]


    γ_ PP^q, q =
      [                   ϵ_ P, L^q q    ϵ_ P, R^q q;    ϵ_ P, L^q q -6 (1 + Q_q^2)              0;    ϵ_ P, R^q q              0 -6 (1 + Q_q^2);                ]   γ_ TP^q, q =
      [            ϵ_P, L^q q ϵ_P, R^q q; ϵ_T, L^q q   - 48 Q_q          0; ϵ_T, R^q q          0     48 Q_q;            ]
         .
 




 



The scalars coefficients at the experimental scale reads : 


    C_S,L^qq(μ_N) =  24 λ^a_Tf_TSQ_qα_e/πlogM/μ_NC_T,L^qq(M) +
    λ^a_S[1-3/2α_e/πlogM/μ_N(1+Q_q^2)]C_S,L^qq(M)





Similarly, the last four diagrams mix  the (pseudo)scalars  to the  tensors. Only the wavefunction diagrams renormalize the tensors, because for the the first and second  diagrams γ^μσγ_μ=0. We obtain  :


    γ_ TT^q, q =
     [                  ϵ_T, L^q q    ϵ_T, R^q q;    ϵ_T, L^q q 2 (1 + Q_q^2)             0;    ϵ_T, R^q q             0 2 (1 + Q_q^2);               ]γ_ S(P)T^q, q =
     [                    ϵ_T, L^q q     ϵ_T, R^q q; ϵ_S (P), L^q q       (-)2 Q_q              0; ϵ_S (P), R^q q              0          2 Q_q;                ]




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   :


    γ_ AV^q, q =
     [            ϵ_V, L^q q ϵ_V, R^q q; ϵ_A, L^q q     12 Q_q          0; ϵ_A, R^q q          0   - 12 Q_q;            ]γ_ VA^q, q =
     [              ϵ_A, L^q q  ϵ_A, R^q q;  ϵ_V, L^q q      12 Q_q           0; ϵ_V
    , R^q q           0    - 12 Q_q;             ]




    C_T,L^qq(μ_N) =  - λ^a_Sf_STQ_qα_e/2πlogM/μ_NC_S,L^qq(M) +
     λ^a_T[1+α_e/2πlogM/μ_N(1+Q_q^2)]C_T,L^qq(M)







 §.§ 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 eqn. (<ref>) and are given in appendix <ref>.











For the decay of light quark (Kaon and D meson decays), the experimental scale is taken as 2 GeV because most of the time, it's the renormalization scale chosen to obtain the lattice form factors.

The evolution of the coefficients (ϵ^e μ ds) involved in the Kaon decays is given by : 





    ( [ ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W = 
    (
    [                         0.631 (0.629)                  0 -214.593 (107.297)                  0                  0                  0;                                     0       1.017(0.991)                  0                  0                  0                  0;                         4.064(-2.032)                  0       1.165(1.167)                  0                  0                  0;                                     0                  0                  0      0.631 (0.629)                  0                  0                   ;                                     0                  0                  0                  0       0.983(1.009)                  0;                                     0                  0                  0                  0                  0       1.165(1.167) ])
    ( [ ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp
 



    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp = 
    (
    [     1.64        0        0        0        0        0  -0.0429        0        0        0;        0        1        0        0        0  0.00857        0        0        0        0;        0        0     1.64        0        0        0        0        0        0   0.0429;        0        0        0        1        0        0        0        0 -0.00857        0;        0        0        0        0     1.64        0   0.0429        0        0        0;        0  0.00857        0        0        0        1        0        0        0        0; -0.00162        0        0        0  0.00162        0    0.849        0        0        0;        0        0        0        0        0        0        0     1.64        0   0.0429;        0        0        0 -0.00857        0        0        0        0        1        0;        0        0  0.00162        0        0        0        0  0.00162        0    0.849;          ])
    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W
 


For the D meson decays, the evolution of the coefficients (ϵ^e μ cu) is given by : 



    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp = 
    (
    [     1.64        0        0        0        0        0   0.0857        0        0        0;        0        1        0        0        0  -0.0171        0        0        0        0;        0        0     1.64        0        0        0        0        0        0  -0.0857;        0        0        0        1        0        0        0        0   0.0171        0;        0        0        0        0     1.64        0  -0.0857        0        0        0;        0  -0.0171        0        0        0        1        0        0        0        0;  0.00325        0        0        0 -0.00325        0    0.847        0        0        0;        0        0        0        0        0        0        0     1.64        0  -0.0857;        0        0        0   0.0171        0        0        0        0        1        0;        0        0 -0.00325        0        0        0        0 -0.00325        0    0.847;          ])
    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W
 



In the B and B_s  meson decay, the reference scale is the b quark mass  (Λ_m_b∼ 4.18 GeV). Thus, the evolution of the coefficients (ϵ^e μ bd and ϵ^e μ bs) is slightly smaller. 

In fact, in eqn. (<ref>), the part with the anomalous dimension  that gives the matrix element in eqn. (<ref>) is multiplied by a factor log(Λ_W/Λ_m_b)/log(Λ_W/Λ_exp) ∼ 0.8. Moreover, the strong coupling constant at Λ_m_b  will also be smaller (α_s(Λ_m_b) ∼ 0.23 and α_s(Λ_exp) ∼ 0.3). Thus, for the B and B_s meson decays, the evolution of the coefficients (ϵ^e μ bd and ϵ^e μ bs ) is given by : 



    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp = 
    (
    [     1.42        0        0        0        0        0  -0.0317        0        0        0;        0        1        0        0        0  0.00686        0        0        0        0;        0        0     1.42        0        0        0        0        0        0   0.0317;        0        0        0        1        0        0        0        0 -0.00686        0;        0        0        0        0     1.42        0   0.0317        0        0        0;        0  0.00686        0        0        0        1        0        0        0        0; -0.00126        0        0        0  0.00126        0    0.890        0        0        0;        0        0        0        0        0        0        0     1.42        0   0.0317;        0        0        0 -0.00686        0        0        0        0        1        0;        0        0  0.00126        0        0        0        0  0.00126        0    0.890;          ])
    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W
 








    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_m_b = 
    (
    [     1.63        0        0        0        0        0  -0.0343        0        0        0;        0        1        0        0        0  0.00686        0        0        0        0;        0        0     1.63        0        0        0        0        0        0   0.0343;        0        0        0        1        0        0        0        0 -0.00686        0;        0        0        0        0     1.63        0   0.0343        0        0        0;        0  0.00686        0        0        0        1        0        0        0        0; -0.00130        0        0        0  0.00130        0    0.849        0        0        0;        0        0        0        0        0        0        0     1.63        0   0.0343;        0        0        0 -0.00686        0        0        0        0        1        0;        0        0  0.00130        0        0        0        0  0.00130        0    0.849;          ])
    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W
 









 §.§ 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
semi-leptonic  branching ratios can both depend on any of the ten
 coefficients, which we arrange in a vector as
 ϵ⃗'⃗= ( [ ϵ_P,L,ϵ_A,L,ϵ_P,R,ϵ_A,R,ϵ_S,L,ϵ_V,L,ϵ_T_L, ϵ_S,R,ϵ_V,R,ϵ_T_R;  ])_Λ_W. The 10 × 10 matrix we need to invert to compute the bounds at Λ_W is now written :


    (M')^-1= ℛ^T
    (
    [ M_2^-1  0_4×6;  0_6×4 M_3^-1;        ])ℛ


where M_2^-1 and M_3^-1 are the 4×4  and 6×6 matrices defined in appendix <ref> we inverted to obtain the bounds at Λ_exp (see section <ref>) and ℛ has the form of  the matrices defined in eqn.  (<ref>), (<ref>) and (<ref>).  Finally, eqn. (<ref>)  is written in the new basis as : 


    ϵ⃗ ⃗'⃗^T(M')^-1ϵ⃗ ⃗'⃗ =1


where ϵ⃗ ⃗'⃗ is the vector of coefficients at Λ_W, (M')^-1 the matrix in eqn. (<ref>)    and the superscript T  means matrix transposition. All the covariance matrices at Λ_W can be found in appendix <ref>. In table <ref>, <ref> and <ref> 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 <ref>. As can be seen in eqn. (<ref>), (<ref>) or (<ref>), the running of the (pseudo)scalar coefficients is ∼ 1.6 (1.4), 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 eqn. (<ref>), (<ref>) and eqn. (<ref>) to (<ref>)) weaken the bounds on pseudoscalar  coefficients at Λ_W  for the Kaon   decay. 
This is due to the fact that the bounds on  ϵ_T^e μ ds  (see the first two columns of table <ref>) are much weaker than the bounds on  ϵ_P^e μ ds  at Λ_exp  (see the   first two columns of the left panel of table <ref>). Thus, the mixing of ϵ_T into ϵ_P will leads 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_P^e μ cu,  ϵ_P^e μ bd and ϵ_P^e μ bs 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 increases a bit  because, similarly to ϵ_P^e μ cu, ϵ_P^e μ bd and ϵ_P^e μ bs, the bounds on all the scalar coefficients (first two columns of the right panel of table <ref>) 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, 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_A^e μ ds and ϵ_V^e μ ds at Λ_exp are very close (see table <ref>), 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 decay. Thus, the bounds on ϵ_A^e μ cu, ϵ_A^e μ bd and ϵ_A^e μ bs do not evolve significantly at Λ_W, but the mixing of the axial into vector coefficients will lead to weaker bounds on ϵ_V^e μ cu, ϵ_V^e μ bd and ϵ_V^e μ bs at Λ_W as shown in the first two columns of the two panels  of   table <ref>. 

 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 <ref>) as for  the bounds on vector and axial coefficients in the Kaon decay (first two columns of table <ref>). 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 <cit.>. 

 
 

 §.§ 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 column of tables <ref>
   to <ref>.
   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 non-zero coefficient
   at a time in the branching ratio (see eqn (<ref>) and (<ref>)). This is different from setting bounds on coefficients (first two columns of table <ref> to <ref>), which are obtained with all coefficients non-zero, 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 column of table  <ref> to <ref>), we still kept only one non-zero coefficients at Λ_exp and considered  only the running of the coefficients (the diagonal terms in eqn. (<ref>) to (<ref>)). For example, computing the sensitivity of the leptonic Kaon decay to a pseudoscalar coefficient  at Λ_W in the SO approximation requires to multiply the first term in eqn.  (<ref>) by the first (or third)  diagonal term squared  in eqn. (<ref>). Then, inverting the product and taking the square root  will give the sensitivity of the decay to the coefficient at  Λ_W. 





 We also computed the bounds in the single operator  (S.O) approximation,  at Λ_exp and Λ_W. This means that we considered one operator at a time and neglected all interferences among operators to compute the bounds at Λ_exp. Then we used the RGEs to evolve the bounds to Λ_W. Moreover, in the S.O approximation, we neglected the mixing between axial and vector operators, because the bounds on ϵ_V and ϵ_A at Λ_exp are close,  which explains why the bounds on ϵ_V and ϵ_A are identical at Λ_exp and Λ_W (see the last two columns of table <ref>).   Neglecting the interferences among operators considerably   simplify the computation of the bounds, because there is no need to inverse a 10 × 10 matrix anymore. For example, to obtain the bound on ϵ_P, we just need to invert the first diagonal term in the matrix of eqn. (<ref>), because the bounds are identical for both left and right chirality of an operator. To compute the bounds at Λ_W on ϵ_S,X, ϵ_P,X and ϵ_T,X, as we neglect the mixing of vector and axial operators, M^-1' in eqn. (<ref>) becomes a product of   3×3 matrices   in the basis ϵ_P,X, ϵ_S,X and ϵ_T,X where X is the chirality (see the last column of table <ref> to <ref>).  But as can be seen in the first and last two columns in table <ref> to <ref>, the S.O approximation gives different bounds compared to the bounds obtained by keeping the interferences. In the case of the semileptonic decays, the bounds obtained on ϵ_S,   ϵ_V and ϵ_T at Λ_exp in the S.O approximation are close to the bounds obtained by keeping the interferences. At Λ_W, only the bounds on ϵ_V are different in the S.O approximation. This can be explained by the fact that the bounds on ϵ_V and ϵ_A at Λ_exp are close in the S.O approximation (see table <ref>), which is not the case when  keeping the interferences. Indeed, in the B and B_s decays, the bounds on ϵ_V at Λ_W are much better in the S.O approximation (table <ref>). This can be seen by comparing the  first two columns in table <ref> to <ref> for   ϵ_S,  ϵ_V and ϵ_T  to the last two columns.

 In the leptonic case, the bounds  obtained on ϵ_A and ϵ_P at Λ_exp and Λ_W in the S.O approximation are better by several orders of magnitude compared to the bounds obtained by keeping the interferences among coefficients, as can bee seen by comparing the first two column and the last two columns of table <ref> and <ref>. This is due to the fact that the interference between axial and pseudoscalar coefficients of different chirality (see eqn. (<ref>)) cannot be neglected. Thus, it is critically important to keep the interferences among operators to compute  correctly the bounds on operator coefficients.
 


 

 §.§ 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 |ϵ| <√( BR^exp)× constant. Thus,
 all the bounds at Λ_exp given in tables <ref> to <ref> can be  updated by  rescaling by √( (BR_new^exp)/(BR_old^exp)) when the data   improves. However, in principle,
 the 10×10 matrix of eqn (<ref>)  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 eqn. (<ref>)). 
   The 6 × 6 matrix in eqn. (<ref>) 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  column  of tables
   <ref> to <ref> (which
   include  the interference terms).

  
 


    In the leptonic decay  (eqn. (<ref>)),
 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 = m_μ in eqn. (<ref>)).
    The other interference terms, proportional to  m_1 = m_e,
    can be neglected. 
    Thus, bounds on  ϵ_A and  ϵ_P at Λ_exp,
    which are a reasonable approximation to the first
    column of tables <ref> and <ref>, can be obtained by
 inverting a  2×2 matrix in the basis ( ϵ_P,X,ϵ_A,Y) where X ∈ L,R and Y ∈ R,L, instead of the 4×4 matrix in eqn. (<ref>).   






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' in eqn. (<ref>)  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 chirality. 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 eqn. (<ref>), and (<ref>) to (<ref>)). 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.


 
 
 In this section, we give details of several approximations that will simplify the computation of the bounds that normally requires to invert large matrices.  For the  semileptonic decay,  neglecting all the interferences between (pseudo)scalar, vector and tensor coefficients (see eqn. (<ref>)) is a reasonable  approximation to compute bounds  on ϵ_S, ϵ_V and ϵ_T at Λ_exp.  Instead of inverting the 6 × 6 matrix in eqn. (<ref>), the computation of the bounds on ϵ_S, ϵ_V and ϵ_T at Λ_exp is much easier and only requires to  invert the three first diagonal terms in the matrix of eqn. (<ref>), because the bounds will be the same for the two chiralities of the same coefficient.  However, to obtain an accurate approximation for the bounds on the coefficients of the leptonic decay  (eqn. (<ref>)) at Λ_exp, it is necessary to keep the interference between axial and pseudoscalar coefficients  of different chirality (with m_2 = m_μ in eqn. (<ref>)), which is stronger than the other interferences (where m_1 = m_e). Thus, computing the bounds on ϵ_P and ϵ_A at Λ_exp  requires to invert a  2×2 matrix in the basis ( ϵ_P,X,ϵ_A,Y) where X ∈ L,R and Y ∈ R,L, instead of the 4×4 matrix in eqn. (<ref>). 

Using these two approximations will give  bounds on  ϵ_S,  ϵ_V and  ϵ_T at Λ_exp , close to those obtained in the first and in the  third column (S.O approximation) of table <ref> to <ref>. Keeping only the interference between ϵ_P,R and ϵ_A,L (or ϵ_P,L and ϵ_A,R) will gives bounds on  ϵ_A and  ϵ_P at Λ_exp  close to those obtained in the first column of table <ref> and <ref> but not close to the sensitivities in the third column (S.O approximation). This means that the interference between ϵ_P,R and ϵ_A,L (or ϵ_P,L and ϵ_A,R) is large and have a strong impact on the computation of the bounds on ϵ_P and ϵ_A . 

In this work, bounds on the coefficients at Λ_exp   can be expressed in the form |ϵ| <√( BR^exp)× constant. Thus, all the bounds at Λ_exp obtained in table <ref> to <ref> can be  rescaled (using the expression  |ϵ| <√( BR^exp)× constant) when the data   improves. 
 

To transport the bounds on ϵ_S,  ϵ_V, ϵ_T, ϵ_A and  ϵ_P at Λ_W in the approximations described above, 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_exp can be evolved to Λ_W by considering M^-1' in eqn. (<ref>)  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 chirality. 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 eqn. (<ref>), and (<ref>) to (<ref>)). 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.


 





§ CONCLUSION


In this paper,

we consider  operators which simultaneously change lepton and quark flavour,
and obtain constraints on the coefficients 
using available data on (semi)leptonic pseudoscalar meson decays.
Section <ref>  lists 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 are included.
The leptonic and semileptonic branching ratios of pseudoscalar mesons, as a function of the operator coefficients, are given in section <ref>.
We find 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 <ref>, <ref> and <ref>
and discussed in  section <ref>. The bounds are obtained
via the appropriate covariance matrices,  which allows to take into account the interferences among operators (see eqn. (<ref>),(<ref>),(<ref>) and (<ref>)). The  matrices are given  in appendix B.
Section <ref> gives  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
 <ref>, <ref> and <ref>.
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 we consider quark-flavor 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 subsection <ref>, 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.     








§ CONSTANTS


 In this section, we give all the constants used in our calculations : 




  P_1   K^0_L     K^+    D^0    D^+     D^+_S     B^0   B^0_S   B^+ 

  f_P_1 (MeV)   155.72 <cit.>     155.6 <cit.>    211.5 <cit.>     212.6 <cit.>    249.8 <cit.>    190.9 <cit.>   230.7<cit.>      187.1 <cit.>

  f_+^P_1 π(0)   0.966 <cit.>      0.966 <cit.>      0.666 <cit.>     0.666 <cit.>   0.666 <cit.>    0.25 <cit.>     0.25 <cit.>    0.25 <cit.> 
 
  f_+^P_1 K(0)   -     -     0.747 <cit.>    0.747 <cit.>   0.747 <cit.>    0.31 <cit.>    0.31 <cit.>   0.31 <cit.>  
 
  λ_+    2.82 × 10^-2 <cit.>    2.97 × 10^-2 <cit.>      -   -   -   -   -   - 
 
  λ_0    1.8 × 10^-2 <cit.>    1.95 × 10^-2 <cit.>       -   -   -   -    -   - 
 
  




All the masses and lifetime can be found in <cit.>.
 


§ KINEMATICS AND FORM FACTORS FOR SEMILEPTONIC DECAYS 

 

In this section, we give the form factor and the detailed scalar product of eqn. (<ref>). 


The  q^2 dependence of the form factors for the Kaon is given by <cit.> : 


    f_+,0^K π(q^2) = f_+^K π(0) ( 1+ λ_+,0q^2/M_π^2);   f_-^K π(q^2) = f_+^K π(0)(λ_0-λ_+)M_K^+^2-M_π^+^2/M_π^+^2


and for the D and B mesons, are given by <cit.> : 

    f_+(q^2) =  f_+(0)/1-q^2/m_1^-^2 ;   f_0(q^2) = f_0(0)/1-q^2/m_0^+^2;    
    f_-(q^2)=(f_0(q^2)-f_+(q^2))M^2-m_3^2/q^2


where λ_+,0 are constants, m_J^P  is the mass of the lightest resonance with the right quantum numbers to mediate the transition (D_s^+ and D_s^*+ for example). We took q^2  = q^2_max =(M - m_3)^2 to compute the form factors f_+, f_- and f_0. All these values can be found in appendix <ref>.







Finally, the scalar product in eqn. (<ref>) can be written as functions of the two kinematical variables q^2 and cosθ <cit.> in the phase space integrals of eqn. (<ref>). 




    p_1.p_2 = q^2  - m_1^2 -m_2^2/2,    p_1.q = q^2 + m_1^2 - m_2^2/2,    p_2.q = q^2  + m_2^2 - m_1^2 /2



    p_3.q = M^2 - m_3^2 -q^2/2,     p_1.p_3 = p_3.q -p_2.p_3,    p_1.P =  p_1.q + 2p_1.p_3,    p_2.P =  p_2.q + 2p_2.p_3



    p_2.p_3 =  1/4q^2
    ( M^2 -m_3^2-q^2  )(q^2  + m_2^2 -m_1^2) + 1/4q^2√(λ(M^2,m_3^2,q^2))√(λ(q^2,m_1^2,m_2^2))cosθ




    k.p_3 = M^2 + m_3^2 - q^2/2,   P.q = M^2 - m_3^2,    P^2 = 2M^2 + 2m_3 - q^2
 




§ RGES



In this section, we give the 10×10 matrices obtained with eqn. (<ref>) we used to obtained the bounds at Λ_W (with eqn. (<ref>)).

For the decay of light quark (Kaon and D meson decays), the experimental scale is taken as 2 GeV because most of the time, it's the renormalization scale chosen to obtain the lattice form factors.



The evolution of the coefficients (ϵ^e μ ds) involved in the Kaon decays is given by : 





    ( [ ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W = 
    (
    [                         0.631 (0.629)                  0 -214.593 (107.297)                  0                  0                  0;                                     0       1.017(0.991)                  0                  0                  0                  0;                         4.064(-2.032)                  0       1.165(1.167)                  0                  0                  0;                                     0                  0                  0      0.631 (0.629)                  0                  0                   ;                                     0                  0                  0                  0       0.983(1.009)                  0;                                     0                  0                  0                  0                  0       1.165(1.167) ])
    ( [ ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp
 



    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp = 
    (
    [     1.64        0        0        0        0        0  -0.0429        0        0        0;        0        1        0        0        0  0.00857        0        0        0        0;        0        0     1.64        0        0        0        0        0        0   0.0429;        0        0        0        1        0        0        0        0 -0.00857        0;        0        0        0        0     1.64        0   0.0429        0        0        0;        0  0.00857        0        0        0        1        0        0        0        0; -0.00162        0        0        0  0.00162        0    0.849        0        0        0;        0        0        0        0        0        0        0     1.64        0   0.0429;        0        0        0 -0.00857        0        0        0        0        1        0;        0        0  0.00162        0        0        0        0  0.00162        0    0.849;          ])
    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W
 


For the D meson decays, the evolution of the coefficients (ϵ^e μ cu) is given by : 



    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp = 
    (
    [     1.64        0        0        0        0        0   0.0857        0        0        0;        0        1        0        0        0  -0.0171        0        0        0        0;        0        0     1.64        0        0        0        0        0        0  -0.0857;        0        0        0        1        0        0        0        0   0.0171        0;        0        0        0        0     1.64        0  -0.0857        0        0        0;        0  -0.0171        0        0        0        1        0        0        0        0;  0.00325        0        0        0 -0.00325        0    0.847        0        0        0;        0        0        0        0        0        0        0     1.64        0  -0.0857;        0        0        0   0.0171        0        0        0        0        1        0;        0        0 -0.00325        0        0        0        0 -0.00325        0    0.847;          ])
    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W
 



In the B and B_s  meson decay, the reference scale is the b quark mass  (Λ_m_b∼ 4.18 GeV). Thus, the evolution of the coefficients (ϵ^e μ bd and ϵ^e μ bs) is slightly smaller. 

In fact, in eqn. (<ref>), the part with the anomalous dimension  that gives the matrix element in eqn. (<ref>) is multiplied by a factor log(Λ_W/Λ_m_b)/log(Λ_W/Λ_exp) ∼ 0.8. Moreover, the strong coupling constant at Λ_m_b  will also be smaller (α_s(Λ_m_b) ∼ 0.23 and α_s(Λ_exp) ∼ 0.3). Thus, for the B and B_s meson decays, the evolution of the coefficients (ϵ^e μ bd and ϵ^e μ bs ) is given by : 



    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_exp = 
    (
    [     1.42        0        0        0        0        0  -0.0317        0        0        0;        0        1        0        0        0  0.00686        0        0        0        0;        0        0     1.42        0        0        0        0        0        0   0.0317;        0        0        0        1        0        0        0        0 -0.00686        0;        0        0        0        0     1.42        0   0.0317        0        0        0;        0  0.00686        0        0        0        1        0        0        0        0; -0.00126        0        0        0  0.00126        0    0.890        0        0        0;        0        0        0        0        0        0        0     1.42        0   0.0317;        0        0        0 -0.00686        0        0        0        0        1        0;        0        0  0.00126        0        0        0        0  0.00126        0    0.890;          ])
    ([ ϵ_P,L; ϵ_A,L; ϵ_P,R; ϵ_A,R; ϵ_S,L; ϵ_V,L; ϵ_T_L; ϵ_S,R; ϵ_V,R; ϵ_T_R;  ])_Λ_W
 





§ COVARIANCES MATRIX



 In this section, we give details of the formalism introduced in section <ref>, eqn. (<ref>). The matrices in the basis    ( ϵ_P,L,ϵ_A,L, ϵ_P,R ,ϵ_A,R) and  (ϵ_S,L ,ϵ_V,L, ϵ_T_L, ϵ_S,R ,ϵ_V,R, ϵ_T_R)  are written : 



    M_2^-1=1/BR_2^exp[
    [                       SP_+' 1/2 SP_+VA_+'  1/2SP_+SP_-'  1/2SP_+VA_-';               1/2 SP_+VA_+'         VA_-'  1/2SP_-VA_+'  1/2VA_+VA_-';                1/2SP_+SP_-'  1/2SP_-VA_+'         SP_-'  1/2SP_-VA_-';                1/2SP_+VA_-'  1/2VA_+VA_-'  1/2SP_-VA_-'         VA_+';               ]]
 



    M_3^-1=1/BR_3^exp[
    [                      SP_+ 1/2 SP_+VA_-   1/2SP_+T_+  1/2SP_+SP_-  1/2SP_+VA_+   1/2SP_+T_-;              1/2 SP_+VA_-         VA_-   1/2VA_-T_+  1/2SP_-VA_-  1/2VA_+VA_-   1/2VA_-T_-;                1/2SP_+T_+   1/2VA_-T_+          T_+   1/2SP_-T_+   1/2VA_+T_+    1/2T_+T_-;               1/2SP_+SP_-  1/2SP_-VA_-   1/2SP_-T_+         SP_- 1/2 SP_-VA_+   1/2SP_-T_-;               1/2SP_+VA_+  1/2VA_+VA_-   1/2VA_+T_+ 1/2 SP_-VA_+         VA_+   1/2VA_+T_-;                1/2SP_+T_-   1/2VA_-T_-    1/2T_+T_-   1/2SP_-T_-   1/2VA_+T_-          T_-;              ]]
 









Inverting M_2^-1 [M_3^-1]  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 limit are not the same for μ^+ e^- and μ^- e^+ in the final state. In this case, we sum the  M_3^-1 for each bound and then invert it to obtain the covariance matrix of section <ref>. The matrix elements of eqn. (<ref>) are written : 


    SP_+' =SP_-'    =  C_2bodyP̃'̃^2 (P_1^2 -m_i^2-m_j^2)
    
    VA_-' =VA_+'   = C_2bodyÃ'̃^2[(P_1^2-m_i^2-m_j^2) (m_i^2+m_j^2) +4m_i^2m_j^2] 
    
    SP_+VA_-'    =SP_-VA_+'= -2C_2bodyP̃'̃Ã'̃
    m_j(P_1^2 +m_i^2 -m_j^2) 
    
    SP_+VA_+'    =SP_-VA_-'= 2C_2bodyP̃'̃Ã'̃
    m_i(P_1^2 +m_j^2 -m_i^2) 
    
    SP_+SP_-'    = -4C_2bodyP̃'̃^2m_jm_i 
    
    VA_+VA_-'    = -4C_2bodyÃ'̃^2P_1^2m_jm_i 
    
    C_2body   =τ_P_1r^*G_F^2/π P_1^2



For simplicity we note d ϕ = ∫_(m_1+m_2)^2^(M-m_3)^2dq^2 ∫_-1^1dcosθ√(λ(M^2,m_3^2,q^2))√(λ(q^2,m_1^2,m_2^2))/q^2 and the matrix elements of eqn.  (<ref>) are written :




    SP_+ = SP_-    = 2C_3bodyS̃^2 (p_1.p_2)d ϕ
    
    VA_+ = VA_-   =1/4C_3body [f_+^2 (
     4(p_1.P)(p_2.P) - 2P^2 (p_1.p_2) ) + f_-^2 (
     4(p_1.q)(p_2.q) - 2q^2 (p_1.p_2) ) 
       + 4f_+f_-   ((p_1.q)(p_2.P) + (p_1.P)(p_2.q)-(p_1.p_2)(P.q)    )] d ϕ
    
    T_+ = T_-    = 4C_3bodyT̃'̃^2 [ 4 (p_1.q)(p_2.P)(P.q) + 4(p_1.P)(p_2.q)(P.q) -2(p_1.p_2)(P.q)^2  
       +2P^2q^2(p_1.p_2) - 4 P^2(p_1.q)(p_2.q) - 4q^2(p_1.P)(p_2.P) 
     ]  d ϕ
    
     SP_+VA_-    = SP_-VA_+ = -2C_3bodyS̃m_2[ (f_+ (p_1.P) + f_-(p_1.q) )] d ϕ
    
     SP_+VA_+    = SP_-VA_- = 2C_3bodyS̃m_1[ (f_+ (p_2.P) + f_-(p_2.q) ) ] d ϕ
    
     SP_+SP_-    = -4 C_3bodyS̃^2 m_1 m_2 d ϕ
    
     VA_+VA_-   =-C_3body m_1 m_2[f_-^2q^2 + f_+^2P^2 + 2 f_+ f_-(P.q)]d ϕ
     T_+T_-   =16C_3bodyT̃'̃^2 m_1 m_2[(P.q)^2-P^2q^2]d ϕ
    
     SP_+T_+    = SP_-T_-= 8C_3bodyS̃T̃[((p_1.P)(p_2.q)-(p_1.q)(p_2.P)  ) ]d ϕ
    
     SP_+T_-    = SP_-T_+=0 
    
    VA_+T_-    = VA_-T_+ =  4C_3bodyT̃'̃m_2[f_+((p_1.q)p^2-(P.p_1)(P.q)) +  f_-((p_1.q)(P.q)-(p_1.P)q^2)]d ϕ
    
    VA_+T_+    = VA_-T_- =  4C_3bodyT̃'̃m_1[ (f_+((P^2)(p_2.q)-(p_2.P)(P.q)) +f_-((p_2.q)(P.q) -(q^2)(p_2.P)) )]d ϕ
    C_3body   =  τ_P_1/π^38G_F^2/512M^3






§ COVARIANCES MATRICES AT Λ_EXP AND Λ_W



In this section, we give the covariance matrix at Λ_exp and at Λ_W, after the RGEs evolution.



 §.§ Kaon decays


Using the upper limit of table <ref>, for the leptonic Kaon decay, we compute the associated covariance matrix in the basis (ϵ_P,L^e μ ds,ϵ_A,L^e μ ds,  ϵ_P,R^e μ dsϵ_A,R^e μ ds) :  



    (
    [  5.38× 10^-14 -2.33× 10^-14 -1.25× 10^-15  1.26× 10^-12; -2.33× 10^-14  2.97× 10^-11  1.26× 10^-12 -4.03× 10^-13; -1.25× 10^-15  1.26× 10^-12  5.38× 10^-14 -2.33× 10^-14;  1.26× 10^-12 -4.03× 10^-13 -2.33× 10^-14  2.97× 10^-11;               ])
 


Then we use the bounds on semileptonic Kaon decay to compute the covariance matrix for the semileptonic decays in the basis (ϵ_S,L^e μ ds,ϵ_V,L^e μ ds , ϵ_T_L^e μ ds, ϵ_S,R^e μ ds,ϵ_V,R^e μ ds, ϵ_T_R^e μ ds)  : 
 




    (
    [ 1.09× 10^-12 3.51× 10^-12 6.11× 10^-12 1.39× 10^-14 1.96× 10^-13 7.49× 10^-13; 3.51× 10^-12 2.44× 10^-11 4.26× 10^-11 1.96× 10^-13 2.10× 10^-12 6.50× 10^-12; 6.11× 10^-12 4.26× 10^-11 1.51× 10^-10 7.49× 10^-13 6.50× 10^-12 1.58× 10^-11; 1.39× 10^-14 1.96× 10^-13 7.49× 10^-13 1.09× 10^-12 3.51× 10^-12 6.11× 10^-12; 1.96× 10^-13 2.10× 10^-12 6.50× 10^-12 3.51× 10^-12 2.44× 10^-11 4.26× 10^-11; 7.49× 10^-13 6.50× 10^-12 1.58× 10^-11 6.11× 10^-12 4.26× 10^-11 1.51× 10^-10;              ])
 


The diagonal elements give the bounds on |ϵ|^2. The bounds on the coefficients are the square root of the  diagonal elements. For instance,  ϵ^e μ d s_S,L is  excluded above √(1.09 × 10^-12).







The covariance matrix in the basis ( [ ϵ_P,L^e μ ds,ϵ_A,L^e μ ds,ϵ_P,R^e μ ds,ϵ_A,R^e μ ds,ϵ_S,L^e μ ds,ϵ_V,L^e μ ds,ϵ_T_L^e μ ds, ϵ_S,R^e μ ds,ϵ_V,R^e μ ds,ϵ_T_R^e μ ds;  ])_Λ_W is :


    (
    [  1.64× 10^-13 -2.55× 10^-14 -1.55× 10^-14  7.73× 10^-13 -2.91× 10^-14  1.31× 10^-12  5.51× 10^-12 -9.15× 10^-16  2.07× 10^-13  5.75× 10^-13; -2.55× 10^-14  2.97× 10^-11  7.73× 10^-13 -4.03× 10^-13 -7.10× 10^-15 -4.64× 10^-13 -4.30× 10^-13  7.35× 10^-16 -2.15× 10^-14 -6.72× 10^-14; -1.55× 10^-14  7.73× 10^-13  1.64× 10^-13 -2.55× 10^-14  9.15× 10^-16 -2.07× 10^-13 -5.75× 10^-13  2.91× 10^-14 -1.31× 10^-12 -5.51× 10^-12;  7.73× 10^-13 -4.03× 10^-13 -2.55× 10^-14  2.97× 10^-11 -7.35× 10^-16  2.15× 10^-14  6.72× 10^-14  7.10× 10^-15  4.64× 10^-13  4.30× 10^-13; -2.91× 10^-14 -7.10× 10^-15  9.15× 10^-16 -7.35× 10^-16  3.22× 10^-13  8.29× 10^-13 -1.11× 10^-12 -8.03× 10^-15 -8.12× 10^-14 -3.49× 10^-14;  1.31× 10^-12 -4.64× 10^-13 -2.07× 10^-13  2.15× 10^-14  8.29× 10^-13  2.44× 10^-11  5.02× 10^-11 -8.12× 10^-14  2.10× 10^-12  7.66× 10^-12;  5.51× 10^-12 -4.30× 10^-13 -5.75× 10^-13  6.72× 10^-14 -1.11× 10^-12  5.02× 10^-11  2.10× 10^-10 -3.49× 10^-14  7.66× 10^-12  2.19× 10^-11; -9.15× 10^-16  7.35× 10^-16  2.91× 10^-14  7.10× 10^-15 -8.03× 10^-15 -8.12× 10^-14 -3.49× 10^-14  3.22× 10^-13  8.29× 10^-13 -1.11× 10^-12;  2.07× 10^-13 -2.15× 10^-14 -1.31× 10^-12  4.64× 10^-13 -8.12× 10^-14  2.10× 10^-12  7.66× 10^-12  8.29× 10^-13  2.44× 10^-11  5.02× 10^-11;  5.75× 10^-13 -6.72× 10^-14 -5.51× 10^-12  4.30× 10^-13 -3.49× 10^-14  7.66× 10^-12  2.19× 10^-11 -1.11× 10^-12  5.02× 10^-11  2.10× 10^-10;               ])




 §.§ D meson meson decays



The bounds of table <ref> on leptonic D meson decay give the following covariance matrix in the basis (ϵ_P,L^e μ cu,ϵ_A,L^e μ cu , ϵ_P,R^e μ cu,ϵ_A,R^e μ cu) : 


    (
    [  3.07× 10^-6 -3.55 ×10^-7 -2.86× 10^-8  7.91× 10^-5; -3.55× 10^-7  2.04 ×10^-3  7.91× 10^-5  7.30× 10^-7; -2.86× 10^-8  7.91× 10^-5  3.07× 10^-6 -3.55× 10^-7;  7.91× 10^-5  7.30× 10^-7 -3.55× 10^-7  2.04× 10^-3;              ])
 



Using bounds on the semileptonic decay of D and D_s meson give  in the basis (ϵ_S,L^e μ cu ,ϵ_V,L^e μ cu,ϵ_T_L^e μ cu,ϵ_S,R^e μ cu,ϵ_V,R^e μ cu,ϵ_T_R^e μ cu) : 
 



    (
    [  1.80× 10^-6  1.32× 10^-7 -3.19× 10^-8 -2.10× 10^-8 -1.61× 10^-7  1.79× 10^-8;  1.32× 10^-7  2.10× 10^-6  3.65× 10^-7 -1.61× 10^-7   9.7× 10^-8  7.06× 10^-7; -3.19× 10^-8  3.65× 10^-7  4.03× 10^-6  1.79× 10^-8  7.06× 10^-7  2.30× 10^-7; -2.10× 10^-8 -1.61× 10^-7  1.79× 10^-8  1.80× 10^-6  1.32× 10^-7 -3.19× 10^-8; -1.61× 10^-7   9.7× 10^-8  7.06× 10^-7  1.32× 10^-7  2.10× 10^-6  3.65× 10^-7;  1.79× 10^-8  7.06× 10^-7  2.30× 10^-7 -3.19× 10^-8  3.65× 10^-7  4.03× 10^-6;              ])





The covariance matrix in the basis ( [ ϵ_P,L^e μ cu,ϵ_A,L^e μ cu,ϵ_P,R^e μ cu,ϵ_A,R^e μ cu,ϵ_S,L^e μ cu,ϵ_V,L^e μ cu,ϵ_T_L^e μ cu, ϵ_S,R^e μ cu,ϵ_V,R^e μ cu,ϵ_T_R^e μ cu;  ])_Λ_W is :




    (
    [  1.15 × 10^-6 -2.16 × 10^-7 -1.15 × 10^-8  4.81 × 10^-5 -1.45 × 10^-8 -2.62 × 10^-8 -2.97 × 10^-7 -1.55 × 10^-9 -8.69 × 10^-7 -1.68 × 10^-8; -2.16 × 10^-7  2.04 × 10^-3  4.81 × 10^-5  7.31 × 10^-7  1.81 × 10^-9  3.50 × 10^-5  8.22 × 10^-9  8.70 × 10^-9 -1.09 × 10^-8  1.99 × 10^-7; -1.15 × 10^-8  4.81 × 10^-5  1.15 × 10^-6 -2.16 × 10^-7  1.55 × 10^-9  8.69 × 10^-7  1.68 × 10^-8  1.45 × 10^-8  2.62 × 10^-8  2.97 × 10^-7;  4.81 × 10^-5  7.31 × 10^-7 -2.16 × 10^-7  2.04 × 10^-3 -8.70 × 10^-9  1.09 × 10^-8 -1.99 × 10^-7 -1.81 × 10^-9 -3.50 × 10^-5 -8.22 × 10^-9; -1.45 × 10^-8  1.81 × 10^-9  1.55 × 10^-9 -8.70 × 10^-9  6.80 × 10^-7  1.03 × 10^-7  2.73 × 10^-7 -5.58 × 10^-9 -5.42 × 10^-8  2.96 × 10^-8; -2.62 × 10^-8  3.50 × 10^-5  8.69 × 10^-7  1.09 × 10^-8  1.03 × 10^-7  2.70 × 10^-6  4.31 × 10^-7 -5.42 × 10^-8  9.66 × 10^-8  8.36 × 10^-7; -2.97 × 10^-7  8.22 × 10^-9  1.68 × 10^-8 -1.99 × 10^-7  2.73 × 10^-7  4.31 × 10^-7  5.62 × 10^-6  2.96 × 10^-8  8.36 × 10^-7  3.21 × 10^-7; -1.55 × 10^-9  8.70 × 10^-9  1.45 × 10^-8 -1.81 × 10^-9 -5.58 × 10^-9 -5.42 × 10^-8  2.96 × 10^-8  6.80 × 10^-7  1.03 × 10^-7  2.73 × 10^-7; -8.69 × 10^-7 -1.09 × 10^-8  2.62 × 10^-8 -3.50 × 10^-5 -5.42 × 10^-8  9.66 × 10^-8  8.36 × 10^-7  1.03 × 10^-7  2.70 × 10^-6  4.31 × 10^-7; -1.68 × 10^-8  1.99 × 10^-7  2.97 × 10^-7 -8.22 × 10^-9  2.96 × 10^-8  8.36 × 10^-7  3.21 × 10^-7  2.73 × 10^-7  4.31 × 10^-7  5.62 × 10^-6;               ])




 §.§ B meson decays

The bound on the leptonic decay of the B meson (see table <ref>) gives the following covariance matrix in the basis (ϵ_P,L^e μ bd,ϵ_A,L^e μ bd,ϵ_P,R^e μ bd,ϵ_A,R^e μ bd)   : 



    (
    [ 5.53 × 10^-8 9.23 × 10^-8 1.20 × 10^-9 3.48 × 10^-6; 9.23 × 10^-8 2.20 × 10^-4 3.48 × 10^-6 6.89 × 10^-6; 1.20 × 10^-9 3.48 × 10^-6 5.53 × 10^-8 9.23 × 10^-8; 3.48 × 10^-6 6.89 × 10^-6 9.23 × 10^-8 2.20 × 10^-4;              ])
 


The covariance matrix in the basis   (ϵ_S,L^e μ bd,ϵ_V,L^e μ bd, ϵ_T_L^e μ bd,ϵ_S,R^e μ bd,ϵ_V,R^e μ bd,ϵ_T_R^e μ bd) is : 
 
 


    (
    [  2.07× 10^-10  1.21× 10^-11  1.52× 10^-12 -3.90× 10^-15 -5.74× 10^-14  5.18× 10^-15;  1.21× 10^-11  2.23× 10^-10  2.81× 10^-11 -5.74× 10^-14  2.87× 10^-14  2.32× 10^-13;  1.52× 10^-12  2.81× 10^-11  4.03× 10^-10  5.18× 10^-15  2.32× 10^-13  3.50× 10^-14; -3.90× 10^-15 -5.74× 10^-14  5.18× 10^-15  2.07× 10^-10  1.21× 10^-11  1.52× 10^-12; -5.74× 10^-14  2.87× 10^-14  2.32× 10^-13  1.21× 10^-11  2.23× 10^-10  2.81× 10^-11;  5.18× 10^-15  2.32× 10^-13  3.50× 10^-14  1.52× 10^-12  2.81× 10^-11  4.03× 10^-10;               ])



The covariance matrix in the basis ( [ ϵ_P,L^e μ bd,ϵ_A,L^e μ bd,ϵ_P,R^e μ bd,ϵ_A,R^e μ bd,ϵ_S,L^e μ bd,ϵ_V,L^e μ bd,ϵ_T_L^e μ bd, ϵ_S,R^e μ bd,ϵ_V,R^e μ bd,ϵ_T_R^e μ bd;  ])_Λ_W is :





    (
    [   2.74 × 10^-8   6.51 × 10^-8  5.94 × 10^-10   2.45 × 10^-6 -1.10 × 10^-12 -4.46 × 10^-10  5.02 × 10^-11  1.89 × 10^-14   1.68 × 10^-8 -8.41 × 10^-13;   6.51 × 10^-8   2.20 × 10^-4   2.45 × 10^-6   6.89 × 10^-6 -2.11 × 10^-12  -1.51 × 10^-6  9.19 × 10^-11  7.76 × 10^-11   4.73 × 10^-8  -3.47 × 10^-9;  5.94 × 10^-10   2.45 × 10^-6   2.74 × 10^-8   6.51 × 10^-8 -1.89 × 10^-14  -1.68 × 10^-8  8.41 × 10^-13  1.10 × 10^-12  4.46 × 10^-10 -5.02 × 10^-11;   2.45 × 10^-6   6.89 × 10^-6   6.51 × 10^-8   2.20 × 10^-4 -7.76 × 10^-11  -4.73 × 10^-8   3.47 × 10^-9  2.11 × 10^-12   1.51 × 10^-6 -9.19 × 10^-11; -1.10 × 10^-12 -2.11 × 10^-12 -1.89 × 10^-14 -7.76 × 10^-11  1.03 × 10^-10  7.83 × 10^-12 -1.03 × 10^-11 -2.10 × 10^-15 -5.78 × 10^-13  3.15 × 10^-15; -4.46 × 10^-10  -1.51 × 10^-6  -1.68 × 10^-8  -4.73 × 10^-8  7.83 × 10^-12   1.06 × 10^-8  3.09 × 10^-11 -5.78 × 10^-13 -3.24 × 10^-10  2.41 × 10^-11;  5.02 × 10^-11  9.19 × 10^-11  8.41 × 10^-13   3.47 × 10^-9 -1.03 × 10^-11  3.09 × 10^-11  5.10 × 10^-10  3.15 × 10^-15  2.41 × 10^-11  4.30 × 10^-14;  1.89 × 10^-14  7.76 × 10^-11  1.10 × 10^-12  2.11 × 10^-12 -2.10 × 10^-15 -5.78 × 10^-13  3.15 × 10^-15  1.03 × 10^-10  7.83 × 10^-12 -1.03 × 10^-11;   1.68 × 10^-8   4.73 × 10^-8  4.46 × 10^-10   1.51 × 10^-6 -5.78 × 10^-13 -3.24 × 10^-10  2.41 × 10^-11  7.83 × 10^-12   1.06 × 10^-8  3.09 × 10^-11; -8.41 × 10^-13  -3.47 × 10^-9 -5.02 × 10^-11 -9.19 × 10^-11  3.15 × 10^-15  2.41 × 10^-11  4.30 × 10^-14 -1.03 × 10^-11  3.09 × 10^-11  5.10 × 10^-10;                ])





 §.§ Bs meson

The bound on the leptonic decay of the B_s meson gives in the basis  (ϵ_P,L^e μ bs,ϵ_A,L^e μ bs,ϵ_P,R^e μ bs,ϵ_A,R^e μ bs) : 
 


    (
    [   3.06 × 10^-8  -1.22 × 10^-8 -3.40 × 10^-10   1.94 × 10^-6;  -1.22 × 10^-8   1.24 × 10^-4   1.94 × 10^-6  -1.80 × 10^-7; -3.40 × 10^-10   1.94 × 10^-6   3.06 × 10^-8  -1.22 × 10^-8;   1.94 × 10^-6  -1.80 × 10^-7  -1.22 × 10^-8   1.24 × 10^-4;                ])
 
 


The bound on the B_s meson  decaying into Kaon (table <ref>) gives in the basis  (ϵ_S,L^e μ bs,ϵ_V,L^e μ bs, ϵ_T_L^e μ bs,ϵ_S,R^e μ bs,ϵ_V,R^e μ bs,ϵ_T_R^e μ bs)  : 
 



    (
    [  5.05 × 10^-10  3.47 × 10^-11  5.07 × 10^-12 -1.13 × 10^-14 -1.65 × 10^-13  1.73 × 10^-14;  3.47 × 10^-11  6.53 × 10^-10  9.54 × 10^-11 -1.65 × 10^-13  8.78 × 10^-14  7.90 × 10^-13;  5.07 × 10^-12  9.54 × 10^-11   1.51 × 10^-9  1.73 × 10^-14  7.90 × 10^-13  1.38 × 10^-13; -1.13 × 10^-14 -1.65 × 10^-13  1.73 × 10^-14  5.05 × 10^-10  3.47 × 10^-11  5.07 × 10^-12; -1.65 × 10^-13  8.78 × 10^-14  7.90 × 10^-13  3.47 × 10^-11  6.53 × 10^-10  9.54 × 10^-11;  1.73 × 10^-14  7.90 × 10^-13  1.38 × 10^-13  5.07 × 10^-12  9.54 × 10^-11   1.51 × 10^-9;                ])


The covariance matrix in the basis ( [ ϵ_P,L^e μ bs,ϵ_A,L^e μ bs,ϵ_P,R^e μ bs,ϵ_A,R^e μ bs,ϵ_S,L^e μ bs,ϵ_V,L^e μ bs,ϵ_T_L^e μ bs, ϵ_S,R^e μ bs,ϵ_V,R^e μ bs,ϵ_T_R^e μ bs;  ])_Λ_W is :





    (
    [   1.52 × 10^-8  -8.62 × 10^-9 -1.69 × 10^-10   1.37 × 10^-6 -1.35 × 10^-12  6.16 × 10^-11  6.41 × 10^-11 -5.11 × 10^-15   9.39 × 10^-9  2.42 × 10^-13;  -8.62 × 10^-9   1.24 × 10^-4   1.37 × 10^-6  -1.80 × 10^-7  1.21 × 10^-13  -8.51 × 10^-7 -1.29 × 10^-11  4.33 × 10^-11  -1.24 × 10^-9  -1.94 × 10^-9; -1.69 × 10^-10   1.37 × 10^-6   1.52 × 10^-8  -8.62 × 10^-9  5.11 × 10^-15  -9.39 × 10^-9 -2.42 × 10^-13  1.35 × 10^-12 -6.16 × 10^-11 -6.41 × 10^-11;   1.37 × 10^-6  -1.80 × 10^-7  -8.62 × 10^-9   1.24 × 10^-4 -4.33 × 10^-11   1.24 × 10^-9   1.94 × 10^-9 -1.21 × 10^-13   8.51 × 10^-7  1.29 × 10^-11; -1.35 × 10^-12  1.21 × 10^-13  5.11 × 10^-15 -4.33 × 10^-11  2.51 × 10^-10  2.21 × 10^-11 -3.90 × 10^-11 -6.11 × 10^-15 -4.33 × 10^-13  9.78 × 10^-15;  6.16 × 10^-11  -8.51 × 10^-7  -9.39 × 10^-9   1.24 × 10^-9  2.21 × 10^-11   6.49 × 10^-9  1.07 × 10^-10 -4.33 × 10^-13  8.57 × 10^-12  1.42 × 10^-11;  6.41 × 10^-11 -1.29 × 10^-11 -2.42 × 10^-13   1.94 × 10^-9 -3.90 × 10^-11  1.07 × 10^-10   1.91 × 10^-9  9.78 × 10^-15  1.42 × 10^-11  1.74 × 10^-13; -5.11 × 10^-15  4.33 × 10^-11  1.35 × 10^-12 -1.21 × 10^-13 -6.11 × 10^-15 -4.33 × 10^-13  9.78 × 10^-15  2.51 × 10^-10  2.21 × 10^-11 -3.90 × 10^-11;   9.39 × 10^-9  -1.24 × 10^-9 -6.16 × 10^-11   8.51 × 10^-7 -4.33 × 10^-13  8.57 × 10^-12  1.42 × 10^-11  2.21 × 10^-11   6.49 × 10^-9  1.07 × 10^-10;  2.42 × 10^-13  -1.94 × 10^-9 -6.41 × 10^-11  1.29 × 10^-11  9.78 × 10^-15  1.42 × 10^-11  1.74 × 10^-13 -3.90 × 10^-11  1.07 × 10^-10   1.91 × 10^-9;                ])
 











    (
    [  3.12× 10^-17  6.03× 10^-16 -3.08× 10^-15  1.16× 10^-13;  6.03× 10^-16  1.93× 10^-15  1.14× 10^-13 -4.29× 10^-12; -3.08× 10^-15  1.14× 10^-13  3.12× 10^-17  6.14× 10^-16;  1.16× 10^-13 -4.29× 10^-12  6.14× 10^-16  2.00× 10^-15;               ])



    (
    [     1.74× 10^-6     4.53× 10^-9     1.76× 10^-8    5.06× 10^-11 7.04×
       10^-10     1.98× 10^-9;     4.53× 10^-9    2.39× 10^-11    4.22× 10^-11    1.22× 10^-13    2.10× 10^-12    7.52× 10^-12;     1.76× 10^-8    4.22× 10^-11    1.82× 10^-10    5.32× 10^-13    7.25× 10^-12    1.97× 10^-11;    5.06× 10^-11    1.22× 10^-13    5.32× 10^-13    4.32× 10^-13    2.23× 10^-12    4.49× 10^-12;    7.04× 10^-10    2.10× 10^-12    7.25× 10^-12    2.23× 10^-12    2.48× 10^-11    5.01× 10^-11;     1.98× 10^-9    7.52× 10^-12    1.97× 10^-11    4.49× 10^-12    5.01× 10^-11    2.06× 10^-10;                 ])



1.74×10^-6     4.53×10^-9     1.76×10^-8     1.82×10^-7     7.04×10^-10     1.81×10^-9 

 4.53×10^-9     2.39×10^-11     4.22×10^-11     6.91×10^-10     2.10×10^-12     7.13×10^-12 

 1.76×10^-8     4.22×10^-11     1.82×10^-10     1.81×10^-9     7.25×10^-12     1.80×10^-11 

 1.82×10^-7     6.91×10^-10     1.81×10^-9     1.74×10^-6     4.61×10^-9     1.76×10^-8 

 7.04×10^-10     2.10×10^-12     7.25×10^-12     4.61×10^-9     2.48×10^-11     4.29×10^-11 

 1.81×10^-9     7.13×10^-12     1.80×10^-11     1.76×10^-8     4.29×10^-11     1.82×10^-10 





 §.§ D meson



    (
    [  1.25× 10^-9  2.60× 10^-8 -1.28× 10^-7  5.15× 10^-6;  2.60× 10^-8  6.90× 10^-9  5.33× 10^-6    -0.000214; -1.28× 10^-7  5.33× 10^-6  1.25× 10^-9  2.51× 10^-8;  5.15× 10^-6    -0.000214  2.51× 10^-8  6.44× 10^-9;              ])



    (
    [        0.186   -0.0000795    -0.000975 -2.44× 10^-6    -0.000149   -0.0000575;   -0.0000795  2.17× 10^-6  9.78× 10^-7 -1.03× 10^-7  9.67× 10^-8  8.36× 10^-7;    -0.000975  9.78× 10^-7  3.49× 10^-5 -4.07× 10^-8  1.66× 10^-7  3.97× 10^-7; -2.44× 10^-6 -1.03× 10^-7 -4.07× 10^-8  7.16× 10^-7  8.20× 10^-8 -2.34× 10^-8;    -0.000149  9.67× 10^-8  1.66× 10^-7  8.20× 10^-8  2.03× 10^-6  4.18× 10^-7;   -0.0000575  8.36× 10^-7  3.97× 10^-7 -2.34× 10^-8  4.18× 10^-7  5.47× 10^-6;              ])



(
[      0.186 -0.0000795  -0.000975     0.0106  -0.000149 -0.0000732; -0.0000795 2.17×10^-6 9.78×10^-7  -0.000154 9.67×10^-8 1.72×10^-7;  -0.000975 9.78×10^-7 3.49×10^-5 -0.0000732 1.66×10^-7 1.35×10^-7;     0.0106  -0.000154 -0.0000732      0.186 -0.0000769  -0.000975;  -0.000149 9.67×10^-8 1.66×10^-7 -0.0000769 2.03×10^-6 9.46×10^-7; -0.0000732 1.72×10^-7 1.35×10^-7  -0.000975 9.46×10^-7 3.49×10^-5;            ]
)




 §.§ B meson




  
ϵ^eμ  bd


    (
    [  5.16× 10^-12  2.58× 10^-10 -5.33× 10^-10   5.40× 10^-8;  2.58× 10^-10  2.06× 10^-11   5.33× 10^-8  -5.39× 10^-6; -5.33× 10^-10   5.33× 10^-8  5.16× 10^-12  2.61× 10^-10;   5.40× 10^-8  -5.39× 10^-6  2.61× 10^-10  2.12× 10^-11;               ])



    (
    [   2.97× 10^-6   2.40× 10^-9   4.00× 10^-8  2.78× 10^-13  2.00× 10^-11  3.51× 10^-12;   2.40× 10^-9  2.20× 10^-10  1.30× 10^-11 -3.57× 10^-14  2.87× 10^-14  2.69× 10^-13;   4.00× 10^-8  1.30× 10^-11   1.09× 10^-9  7.77× 10^-15  3.67× 10^-13  3.79× 10^-14;  2.78× 10^-13 -3.57× 10^-14  7.77× 10^-15  8.12× 10^-11  7.64× 10^-12  1.12× 10^-12;  2.00× 10^-11  2.87× 10^-14  3.67× 10^-13  7.64× 10^-12  2.26× 10^-10  3.30× 10^-11;  3.51× 10^-12  2.69× 10^-13  3.79× 10^-14  1.12× 10^-12  3.30× 10^-11  5.50× 10^-10;               ])



(
[  2.97×10^-6  2.40×10^-9  4.00×10^-8 2.58×10^-10 2.00×10^-11 2.79×10^-12;  2.40×10^-9 2.20×10^-10 1.30×10^-11 1.98×10^-11 2.87×10^-14 3.62×10^-13;  4.00×10^-8 1.30×10^-11  1.09×10^-9 2.79×10^-12 3.67×10^-13 1.77×10^-14; 2.58×10^-10 1.98×10^-11 2.79×10^-12  2.97×10^-6  2.43×10^-9  4.00×10^-8; 2.00×10^-11 2.87×10^-14 3.67×10^-13  2.43×10^-9 2.26×10^-10 1.32×10^-11; 2.79×10^-12 3.62×10^-13 1.77×10^-14  4.00×10^-8 1.32×10^-11  1.09×10^-9;             ]
)





  
ϵ^eμ  bs


    (
    [ 1.34× 10^-11  1.40× 10^-7 -1.39× 10^-9 6.88× 10^-10;  1.40× 10^-7 5.31× 10^-11 6.79× 10^-10   -0.0000144; -1.39× 10^-9 6.79× 10^-10 1.34× 10^-11  1.42× 10^-7; 6.88× 10^-10   -0.0000144  1.42× 10^-7 5.46× 10^-11;              ])



    (
    [     0.0000113  4.18× 10^-11   1.55× 10^-7  8.64× 10^-13   1.41× 10^-8  1.44× 10^-11;  4.18× 10^-11  6.55× 10^-10  2.98× 10^-13 -2.20× 10^-11  9.07× 10^-14  1.89× 10^-10;   1.55× 10^-7  2.98× 10^-13    3.47× 10^9  2.36× 10^-14  2.49× 10^-10  1.65× 10^-13;  8.64× 10^-13 -2.20× 10^-11  2.36× 10^-14  1.99× 10^-10  1.06× 10^-13 -6.33× 10^-12;   1.41× 10^-8  9.07× 10^-14  2.49× 10^-10  1.06× 10^-13  6.74× 10^-10  5.76× 10^-13;  1.44× 10^-11  1.89× 10^-10  1.65× 10^-13 -6.33× 10^-12  5.76× 10^-13    2.10× 10^9;               ])




(
[ 1.13 ×10^-5 4.18×10^-11  1.55×10^-7  1.06×10^-9  1.41×10^-8 1.22×10^-11; 4.18×10^-11 6.55×10^-10 2.98×10^-13  1.39×10^-8 9.07×10^-14 2.46×10^-10;  1.55×10^-7 2.98×10^-13  3.47×10^-9 1.22×10^-11 2.49×10^-10 1.04×10^-13;  1.06×10^-9  1.39×10^-8 1.22×10^-11 1.13 ×10^-5 4.24×10^-11  1.55×10^-7;  1.41×10^-8 9.07×10^-14 2.49×10^-10 4.24×10^-11 6.74×10^-10 3.02×10^-13; 1.22×10^-11 2.46×10^-10 1.04×10^-13  1.55×10^-7 3.02×10^-13  3.47×10^-9;             ]
)





The evolutions of the coefficients are thus :


    ϵ^l_1l_2q_iq_j_S,XY (Λ_W) = 
    λ^a_S[ 1 - 3α_e/2π(1+Q_q^2) log( Λ_exp/Λ_W)  ]_{[ 0.629   (Q_q = -1/3); 0.631    (Q_q = 2/3) ]. ϵ^l_1l_2q_iq_j_S,XY (Λ_exp) - 
    24λ^a_Tf_TSα_e/πQ_q log( Λ_exp/Λ_W)  _{[ 107.297    (Q_q = -1/3);  214.593    (Q_q = 2/3) ]. ϵ^l_1l_2q_iq_j_T_R,L (Λ_exp)



    ϵ^l_1l_2q_iq_j_T_R,L (Λ_W) = 
    λ^a_T[ 1 + α_e/2π(1+Q_q^2) log( Λ_exp/Λ_W)  ]_{[ 1.167    (Q_q = -1/3);  4.064    (Q_q = 2/3) ]. ϵ^l_1l_2q_iq_j_T_R,L (Λ_exp)-
    λ^a_Sf_STα_e/2π(1+Q_q^2) log( Λ_exp/Λ_W)  _{[    2.032   (Q_q = -1/3); 1.16549     (Q_q = 2/3) ]. ϵ^l_1l_2q_iq_j_S,XY (Λ_exp)



    ϵ^l_1l_2q_iq_j_V,LL (Λ_W) = 
    [ 1 - 3 α_e/πQ_qlog( Λ_exp/Λ_W)  ]_{[ 0.991   (Q_q = -1/3); 1.017    (Q_q = 2/3) ]. ϵ^l_1l_2q_iq_j_V,LL (Λ_exp);   ϵ^l_1l_2q_iq_j_V,LL (Λ_W) = 
    [ 1 + 3 α_e/πQ_qlog( Λ_exp/Λ_W)  ]_{[ 1.009   (Q_q = -1/3); 0.983    (Q_q = 2/3) ]. ϵ^l_1l_2q_iq_j_V∓ A (Λ_exp)




  §.§.§ Kaon decay  ϵ^e μ ds from ds→ e^±μ^∓ and s→d e^±μ^∓




    ϵ^e μ  ds_S,XY (Λ_W) = ;  ϵ^e μ  ds_T_R,L (Λ_W) = 
       ϵ^e μ  ds_V,LL (Λ_W) = (P_XP_X);   ϵ^e μ  ds_V,LL (Λ_W) = (P_XP_Y)




  §.§.§ D meson decay : ϵ^e μ cu from cu→ e^±μ^∓and  c→ u e^±μ^∓, 



    ϵ^e μ  cu_S,XY (Λ_W) = ;  ϵ^e μ  cu_T_R,L (Λ_W) = 
       ϵ^e μ  cu_V,LL (Λ_W) = (P_XP_X);   ϵ^e μ  cu_V,LL (Λ_W) = (P_XP_Y)




  §.§.§ B meson decay : ϵ^e μ bd from db→ e^±μ^∓and  b→ d e^±μ^∓
 ϵ^e μ bs from c→ u e^±μ^∓



    ϵ^e μ  bd_S,XY (Λ_W) = ;  ϵ^e μ  bd_T_R,L (Λ_W) = 
       ϵ^e μ  bd_V,LL (Λ_W) = (P_XP_X);   ϵ^e μ  bd_V,LL (Λ_W) = (P_XP_Y)



    ϵ^e μ  bs_S,XY (Λ_W) = ;  ϵ^e μ  bs_T_R,L (Λ_W) = 
       ϵ^e μ  bs_V,LL (Λ_W) = (P_XP_X);   ϵ^e μ  bs_V,LL (Λ_W) = (P_XP_Y)










ieeetr 


