The decays $Σ^{+}\to p \ell^{+}\ell^{-}$ within the standard model and beyond
Arnab Roy Jusak Tandean German Valencia

=17pt =5pt




arnab.roy1@monash.edu
School of Physics and Astronomy, Monash University, Wellington Road, Clayton, Victoria 3800, Australia


jtandean@yahoo.com
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China


german.valencia@monash.edu
School of Physics and Astronomy, Monash University, Wellington Road, Clayton, Victoria 3800, Australia



 

Motivated by the LHCb measurement of the hyperon decay mode Σ^+→ pμ^+μ^- and prospects for improvement, we revisit the estimates for the rate and forward-backward asymmetry within the standard model and beyond. 
The standard model prediction has a fourfold ambiguity, and we suggest ways to resolve it with other measurements, including possible studies of Σ^+→ p e^+e^- in the BESIII and LHCb experiments. 
We use the recent BESIII measurements of Σ^+→ p γ and Σ^+→ pπ^0 to reduce the uncertainty in the long-distance contribution to Σ^+→ pμ^+μ^-. 
Beyond the standard model, we consider a general effective Hamiltonian at low energy with ten operators whose Wilson coefficients parametrize the new physics. 
We derive expressions for the Σ^+→ pμ^+μ^- rate and the associated muon forward-backward asymmetry in terms of these coefficients. 
Finally, we implement the existing constraints from kaon decays, pointing out the extent to which this hyperon mode can constrain the directions in parameter space to which the kaons are not sensitive.



PACS numbers: 

The decays Σ^+→ p ℓ^+ℓ^- within the standard model and beyond
    German Valencia
    April 29, 2024
=============================================================







§ INTRODUCTION

  
The LHCb Collaboration published a new measurement of the rate of the rare hyperon decay mode  Σ^+→ pμ^+μ^-  in 2018 <cit.> and is expected to present an improved analysis with more data soon. In combination with the renewed interest in future hyperon and kaon experiments <cit.>, this motivates us to revisit the theoretical status of this mode both within the standard model (SM) and beyond. 

In the SM the decays  Σ^+→ pℓ^+ℓ^-  with  ℓ=e,μ  each receive short-distance (SD) and long-distance (LD) contributions. The former are known to be significantly smaller than the latter <cit.>. 
We take another look at the SD component for completeness, as previous calculations have partly relied on the value of a Wilson coefficient that predates the discovery of the top quark <cit.>. 
We update this number with the formalism of Ref. <cit.>, arriving at a larger value than what has been quoted before, but the corresponding total SD amplitude is still substantially below its LD counterpart.

Although the LD contributions dominate the rate, they suffer from multiple uncertainties, which include a fourfold ambiguity in the prediction <cit.>. 
Addressing this is one of the goals of the present paper. Critical input to this examination is the measurement of the rate of the radiative channel  Σ^+→ pγ,  where there is a recent precise result from the BESIII experiment <cit.>. 
We employ this information, along with a new fit to the nonleptonic decays  Σ^+→ nπ^+,pπ^0  prompted by another recent BESIII data <cit.>, to reduce the uncertainty in the LD form-factors that derives from experimental input. 
The fourfold ambiguity in the SM estimation remains, and we propose possible measurements to resolve or alleviate this issue. 
The theoretical uncertainty in the LD calculation is partially quantified by considering extraction of the form factors in relativistic baryon chiral perturbation theory and in its heavy-baryon formulation.

After the complete SM treatment of  Σ^+→ pℓ^+ℓ^-,  we predict in the  ℓ=μ  case the rate, the dimuon spectrum, the muon forward-backward asymmetry, and the additional observable f_L^ in the angular distribution, as well as the the  ℓ=e  rate. 
The results are presented for the four solutions of the form factors that can be used for comparison with future experimental measurements. 

We also consider the constraints that the  ℓ=μ  mode can place on new physics (NP) parametrized in terms of ten Wilson coefficients in the low-energy effective Hamiltonian and how they are complementary to those arising from kaon decay modes. 
Even though the restrictions from the kaon sector are much stronger when they exist, we identify the directions in parameter space to which the kaon measurements are blind and constrain them with the existing information on the Σ^+→ pμ^+μ^- rate.

In Secs.  <ref> and <ref> we deal respectively with the SD and LD amplitudes for  Σ^+→ pℓ^+ℓ^-  in the SM, and in Sec.  <ref> evaluate the corresponding rates and other observables.
In Sec. <ref> we look at the impact of potential NP manifesting itself via ten low-energy effective  s→ dℓ^+ℓ^-  operators. 
In Sec. <ref> we address the effects of the NP on various kaon observables. 
In Sec. <ref> we discuss the complementarity of the hyperon and kaon sectors in probing the NP.  
We present our conclusions in Sec. <ref>.



§ SM SHORT-DISTANCE CONTRIBUTIONS


In the SM, the SD transition  s→ dℓ^+ℓ^-  arises from Z-penguin, box, and photon-penguin diagrams <cit.>, which involve internal up-type quarks and W-boson. 
At QCD renormalization scales of order 1 GeV, this is described by the effective Hamiltonian <cit.>

    H_ eff^sm    = G_ F^/√(2) V_ud^* V_us^ [ (z_7V^+τ y_7V^) O_7V^+ τ y_7A^ O_7A^ + C_7γ^ O_7γ^]  ,

where G_ F is the Fermi coupling constant, V_kl denotes an element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix,
z_7V^, y_7V,7A^, and C_7γ are the Wilson coefficients,
τ =- V_td^*V_ts^/(V_ud^*V_us^),   

    O_7V^    =  2 dγ^κ P_L^ s ℓγ_κ^ℓ , 
    O_7A^ =  2 dγ^κ  P_L^ s ℓγ_κ^γ_5^ℓ ,
    
    
    O_7γ^    =  e/8π^2 dσ^κν( m_s^ P_R^ + m_d^  P_L^) s F_κν^ ,  P_L,R^ = 12(1∓γ_5^)  ,

with e being the proton's electric charge, m_d(s) the d(s)-quark mass, and F_κν^ the photon field-strength tensor.
The total SD contribution to the amplitude for  Σ^+→ pℓ^+ℓ^-  is then 

    M_ SM^ SD    = ⟨ pℓ^+ℓ^-| H_ eff^sm|Σ^+⟩
        = G_ F^V_ud^*V_us^/√(2)[ {⟨ p|dγ^κ(1-γ_5^)s|Σ^+⟩[ (z_7V^+τ y_7V^) u̅_ℓ^γ_κ^v_ℓ̅^ + τ y_7A^ u̅_ℓ^γ_κ^γ_5^v_ℓ̅^];              - i  e^2 C_7γ m_s^/8π^2q^2⟨ p|dσ^νκ(1+γ_5^)s|Σ^+⟩ q_κ^ u̅_ℓ^γ_ν^v_ℓ̅^} , ]

where u_ℓ^ (v_ℓ̅) represents the Dirac spinor of the emitted (anti)lepton, terms linear in m_d have been dropped from the last line because  m_d≪ m_s,  and  q=P_Σ^-P_p  is the difference between the Σ^+ and proton momenta.  

Dealing with the decay rate from M_ SM^ SD requires knowing the hadronic matrix elements in it.
From the leading-order strong Lagrangian in chiral perturbation theory (χPT), one derives <cit.> 

    ⟨ p|dγ^κ s|Σ^+⟩    =  -u̅_p^γ^κ u_Σ^ ,    
    ⟨ p|dγ^κγ_5^s|Σ^+⟩    =  (D-F) (
    u̅_p^γ^κγ_5^ u_Σ^ + m_Σ+m_p/q^2-m_K^2  q^κ u̅_p^γ_5^u_Σ^)  ,

where u_p,Σ (m_p,Σ) denote the Dirac spinors (masses) of the baryons, D and F are parameters which can be determined from the measurements on hyperon semileptonic decays <cit.>, and m_K is the average kaon mass. 
For the tensor currents, one can express[Although ⟨ p|dσ^κνs|Σ^+⟩ involves four form-factors which are functions of q^2, after contracting it with ε_κ^* q_ν^ and imposing  ε· q=0  only one combination of them remains at q^2=0, as Eq. (<ref>) indicates.
The matrix element of  dσ^κνγ_5^s is related to that of  dσ^κνs  because of the identity
 2iσ^κνγ_5^=ϵ^κνφςσ_φς^.]

    ⟨ p|dσ^κν(1,γ_5^)s|Σ^+⟩ε_κ^*q_ν^    =  c_σ^ u̅_p^σ^κν(1,γ_5^)u_Σ^ ε_κ^*q_ν^ ,

where ε stands for the polarization four-vector of an on-shell photon and c_σ^ is a constant.

Numerically, we adopt the CKM parameters available from Ref. <cit.>, the Wilson coefficients typically given in the literature, namely
 z_7V^ = -0.046α_ e^,   y_7V^ = 0.73α_ e^,  y_7A^ = -0.68α_ e^  <cit.>, and  C_7γ=0.72  <cit.>,[This is nearly real, results mainly from O_7γ mixing with  O_2=dγ^ν P_L^u uγ_ν^P_L^s  under QCD renormalization <cit.>, and is hugely enhanced relative to the value without QCD corrections,  C_7γ=(-7.1-2.5i)×10^-4,  which is dominated by the top contribution, the charm and up ones being four times smaller and negligible, respectively.] and  m_s^=123 MeV,  all evaluated at a renormalization scale of 1  GeV, with  α_ e^ = α_ e^(m_Z) = 1/127.95 <cit.>. 
For the Σ^+ lifetime and hadron masses, we employ the ones supplied by Ref. <cit.>.
We also have  c_σ^=0.3  from a quark-model computation <cit.> and  D = 0.81±0.01  and  F = 0.47±0.01  from fitting to hyperon semileptonic decay data <cit.>.

From the foregoing, the SD amplitude induced by O_7γ (O_7V and O_7A) alone yields the branching fractions  B(Σ^+→ pe^+e^-)_ SD^ = 2.3 (0.25)×10^-10  and  B(Σ^+→ pμ^+μ^-)_ SD^=1.1 (1.1)×10^-12.   
They change to  B(Σ^+→ pe^+e^-)_ SD^ = 2.4×10^-10  and  B(Σ^+→ pμ^+μ^-)_ SD^ = 6.0×10^-13  if O_7γ and O_7V,7A are present simultaneously, indicating partial cancellation among their contributions in the muon case.
These numbers are roughly five orders of magnitude smaller than their experimental counterparts <cit.>.



§ SM LONG-DISTANCE CONTRIBUTIONS


The SD contribution being relatively tiny, the dominant component of  Σ^+→ pℓ^+ℓ^-  has been estimated to be the long-distance photon-mediated process  Σ^+→ pγ^*→ pℓ^+ℓ^-.   
If the photon is on-shell, the amplitude is parametrized by two form factors <cit.>, a and b, that can be partly determined via the radiative weak decay  Σ^+→ pγ.  
For the latter mode, the effective Lagrangian has the form

    L_Σ^+→ pγ^    =  eG_ F/2 p( a +γ_5^ b ) σ^κνΣ^+ F_κν ,

implying a partial width Γ_γ and distribution dΓ_γ/d(cosϑ) given by 

    Γ_γ    =  e^2 G_ F^2/π(|a|^2+|b|^2) E_γ^3  , 
    
    dΓ_γ/d(cosϑ)    ∼  1+α_γ cosϑ , α_γ = 2  Re (ab^*)/|a|^2+|b|^2 ,

where E_γ denotes the photon's energy in the rest frame of the Σ^+ and ϑ is the angle between its polarization and the proton's three-momentum in this frame <cit.>.

When the photon is virtual, the amplitude for  Σ^+→ pγ^*  contains two additional form-factors <cit.>, c and d, and is expressible as

    M(Σ^+→ pγ^*)     = 
    -  eG_ F^ u̅_p^[ iσ^κν q_κ^(a+γ_5^b) + ( γ^ν q^2 -  q  q^ν) (c+γ_5^d) ] u_Σ^ ε_ν^*  ,

where q stands for the photon's four-momentum.
Its contribution to  Σ^+→ pℓ^+ℓ^-  is then found to be

    M_ SM^ LD    =  e^2 G_ F^[ -i/q^2 u̅_p^(a+γ_5^b) σ^κν q_κ^ u_Σ^ - u̅_p^γ^ν(c+γ_5^d) u_Σ^] u̅_ℓ^γ_ν^ v_ℓ̅^ .

In general  a, b, c, and d depend on q^2 and can be complex, and the first two are constrained at  q^2=0  by the data in the real-photon case, to which we turn next.

There is experimental information available on the branching fraction ℬ_γ of  Σ^+→ pγ  and its decay asymmetry parameter α_γ, mostly gathered decades ago <cit.>.
Last year the BESIII Collaboration <cit.> reported fresh data on these quantities:

    ℬ_γ    = (0.996± 0.021_ stat±0.018_ syst)×10^-3 ,
    α_
    γ    =  -0.652 ± 0.056_ stat±0.020_ syst .

The former is less than the corresponding PDG value of  (1.23±0.05)× 10^-3 <cit.> by 4.2 standard deviations, whereas the α_
γ finding is compatible with the prior measurements <cit.>. 
Accordingly, for numerical work, we may simply adopt the above BESIII number for ℬ_γ, while for α_γ we can take the average of the BESIII and earlier results, yielding  α_
γ=-0.694±0.047.  
Putting together these choices with Eq. (<ref>) and employing the hadron masses and Σ^+ lifetime from Ref. <cit.>, we then arrive at the  q^2=0  combinations

    |a(0)|^2+|b(0)|^2    =  (13.48± 0.19)^2 MeV^2 , 
     Re(a(0) b^*(0))     =  (-63.1±4.6)  MeV^2 ,

having additionally used  e^2 = 4πα_ e^(0),  with  α_ e(0)=1/137.036  <cit.>, in Γ_γ.
The LHCb experiment may be able to measure ℬ_γ and α_γ as well as they have measured similar quantities in the b-baryon sector <cit.>.

As elaborated in Ref. <cit.>, to estimate how a, b, c, and d depend on q^2, their imaginary parts are first computed with the aid of unitarity arguments, and among the main inputs are the empirical S- and P-wave amplitudes for the nonleptonic channels  Σ^+→ nπ^+,pπ^0.  
Subsequently, with the resulting Im(a,b) evaluated at  q^2=0,  the real parts of a and b are approximated to be constant and fixed from the observed ℬ_
γ and α_
γ values of  Σ^+→ pγ,  up to a fourfold ambiguity.
Since there are new relevant data on  Σ^+→ pπ^0  from BESIII <cit.>, this procedure needs to be updated. 
Thus, writing the amplitude for  Σ^+→ Nπ  as  M_Σ^+→ Nπ = iG_ F^m_π^+^2 u̅_N (A_Nπ - γ_5B_Nπ) u_Σ  and neglecting the small phases of A_Nπ and B_Nπ, we arrive at[We have used the branching fractions  ℬ(Σ^+→ nπ^+)=(48.31±0.30)%  and  ℬ(Σ^+→ pπ^0)=(51.57±0.30)% and decay asymmetry parameter  α_π(Σ^+→ nπ^+)=0.068±0.013  from Ref. <cit.>, based on measurements done over 40 years ago, as well as  α_π(Σ^+→ pπ^0)=-0.986±0.015  from averaging the recent BESIII findings on α_π for Σ^+→ pπ^0 and its antiparticle partner <cit.> and the corresponding earlier measurements <cit.>.] 

    A_nπ^+    =  0.062±0.012  ,     B_nπ^+    =  18.534±0.066  ,    
    
    A_pπ^0    =  -1.413±0.054  ,     B_pπ^0    =  11.97±0.64  .

With these, we obtain  Ima(0) = (2.881
±0.063) MeV  and  Imb(0) = (-1.855±0.092) MeV,  which lead to the four solutions collected in the upper half of Table <ref>. 
Employing the heavy-baryon formalism instead, we find  Ima(0) = (6.98±0.10) MeV  and  Imb(0) = (-0.56±0.11) MeV,  and hence another four solutions placed in the lower half of Table <ref>. 
In all of our figures we will distinguish between the solutions listed in this table with this color coding: blue for solution 1, yellow solution 2, green solution 3, and red solution 4.


 

It is interesting that, as pointed out a while ago <cit.>, the parameter

    γ_γ    = |a(0)|^2-|b(0)|^2/|a(0)|^2+|b(0)|^2

may be determined in a weak radiative hyperon decay if the linear polarization of the emitted photon and the spins of the initial and final baryons can all be detected. 
If so, γ_γ could reduce the fourfold ambiguity to a twofold ambiguity. 
For the situation illustrated by Table <ref>, from the top (bottom) half we get γ_γ=0.71±0.04  for solutions 1 and 4 and γ_γ=-0.66±0.04  for solutions 2 and 3 (γ_γ=0.60±0.09  for solutions 1 and 4 and  γ_γ=-0.07±0.10  for solutions 2 and 3). 
We see that just a measurement of the sign of γ_γ may suffice to decrease the ambiguity to twofold.



§ DECAY RATES AND OBSERVABLES


The different contributions to  Σ^+→ pℓ^+ℓ^-  can be  incorporated into the general expression for the amplitude <cit.> 

    M =     [ i q_κ^ u̅_p^( ã
    + γ_5^b̃) σ^νκ u_Σ^
    - u̅_p^γ^ν(c̃+γ_5^d̃) u_Σ^] u̅_ℓ^γ_ν^ v_ℓ̅^ + u̅_p^γ^ν(ẽ+γ_5^f̃) u_Σ^ u̅_ℓ^γ_ν^γ_5^ v_ℓ̅^
        + u̅_p^(g̃+γ_5^h̃)u_Σ^ u̅_ℓ^v_ℓ̅^ + u̅_p^(j̃+γ_5^k̃)u_Σ^ u̅_ℓ^γ_5^v_ℓ̅^ ,


where  q=P_Σ^-P_p=P_+^+P_-^,  with P_±^ denoting the four-momenta of ℓ^±. 
The parameters   ã,b̃,...,k̃  in Eq. (<ref>) may contain both  SM and NP contributions.
The formula for the rate derived from M can be found in Ref. <cit.> and hence will not be reproduced here. 

From Secs.  <ref> and <ref>, the SD and LD amplitudes in the SM yield


    ã    = G_ F^/q^2( a  e^2 - λ_u^*/√(2)c_σ^ C_7γ^ e^2 m_s^/8π^2)  ,    b̃    = G_ F^/q^2( b  e^2 - λ_u^*/√(2)c_σ^ C_7γ^ e^2 m_s^/8π^2)  ,
    
    c̃    =  G_ F^( c  e^2 + λ_u^*z_7V^-λ_t^*y_7V^/√(2))  ,    d̃    =  G_ F^[ d  e^2 + (D-F) λ_u^*z_7V^-λ_t^*y_7V^/√(2)]  ,
    
    ẽ    = G_ F^/√(2) λ_t^*y_7A^ ,    f̃    = D-F/√(2)  G_ F^ λ_t^* y_7A^ ,    
    k̃    = √(2) (D-F) G_ F^ λ_t^* y_7A^  m_ℓ^ m_Σ^+m_p^/q^2-m_K^2 .

These lead to the SM predictions for the branching fractions collected in Table <ref>. 
The error quoted in the third column, under B, is exclusively due to the quantities in Eqs. (<ref>) and (<ref>). 
The calculation of the real parts of c and d is the most model-dependent, and one can compare the entries in the third and fourth columns to get an indication of the model-dependent contribution to the rates. 
Parametric uncertainty in other quantities entering the rates is much smaller, and we ignore it. 
Finally, we can estimate the theoretical uncertainty arising from the extraction of the imaginary part of the form factors at  q^2=0  by comparing the results obtained with relativistic or heavy-baryon χPT. 
The range of predictions obtained by interpolating between these two sets of form factors is displayed in Fig. <ref>.



Measurements of the branching fraction of  Σ^+→ p e^+e^-  may offer an additional way to resolve the fourfold ambiguity. 
In this mode, the dominance of long-distance contributions is more pronounced, owing to the fact that the a and b terms in the amplitude are much enhanced at low values of the dielectron invariant mass,  M_ee=√(q^2)≥2m_e^.  
As a consequence, the predictions for the branching fraction  B_ee≡ B(Σ^+→ p e^+e^-) corresponding to the various solutions are similar to one another, as can be seen in the sixth column of Table <ref>. 
Nevertheless, the difference between solutions can be magnified by raising the minimum of M_ee. 
This might be achieved by applying, for instance, the cut  M_ee≥ 100 MeV,  leading to the numbers in the last column of Table <ref>.
Thus, the selection of an appropriate lower bound on M_ee could make the predictions of the different solutions more distinguishable. 
This point is further illustrated by Fig. <ref>.
Although the implementation of the M_ee min cut has a price, which is a decreased rate, the latter may still be within the sensitivity reach of BESIII <cit.> and LHCb <cit.>, depending on the choice of M_ee min.



It is worth remarking here that the range of the B_ee results in the sixth column of Table <ref> is roughly 20% less than the corresponding predictions made in Refs. <cit.>.  
This is largely because of our adoption of the BESIII finding of ℬ_γ quoted in Eq. (<ref>), which is also  ∼ 20% lower than the present PDG average <cit.>. 
It goes without saying that the upcoming measurements of  Σ^+→ p e^+e^-  by BESIII and LHCb can be anticipated to provide us with highly desirable pertinent information.

Returning to the  ℓ=μ  case, one can write the double differential distribution <cit.>

    d^2Γ(Σ^+→ p μ^+μ^-)/dq^2  dcosθ_μ    = [ 3/8(1+cos^2θ_μ) (1-f_L^) +  A_ FB^cosθ_μ + 3/4  f_L^sin^2θ_μ] Γ'  ,

where θ_μ stands for the angle between the directions of the μ^- and p in the rest frame of the μ^+μ^- system, f_L^ is the fraction of longitudinally polarized dimuons, A_ FB designates the muon forward-backward asymmetry, and  Γ'≡ dΓ(Σ^+→ p μ^+μ^-)/dq^2.   In terms of the general coefficients in M above, we have <cit.>

    A_ FB^ = β^2 λ̅/64π^3  Γ'  m_Σ^3  Re[ {[ _+^ã^*f̃
    - _-^b̃^* ẽ
    - ( ã^*g̃+b̃^*h̃) m_μ
    + c̃^*f̃ + d^*ẽ] q^2;                                - ( _+^c̃^* g̃
    - _-^d̃^* h̃) m_μ} , ]


where 


    β    = √(1-4m_μ^2/q^2) ,    λ̅    = ( M_+^2-q^2) ( M_-^2-q^2)  ,     M_±^    =  m_Σ^± m_p^ .


It is also of interest to have the integrated forward-backward asymmetry 


    Ã_ FB^    = 1/Γ(Σ^+→ pμ^+μ^-)∫Γ'  A_ FB^  dq^2  .

The SM prediction for it can be viewed in Table <ref>, normalized by the respective rate for each solution, and its differential distribution is depicted in Fig. <ref>.
Evidently, there is a sign flip for two of the solutions (1 and 2 in both the relativistic and heavy-baryon cases), namely, dÃ_ FB/dM_μμ is negative.




Also consequential is the dimuon spectrum,
    dΓ̂/dM_μμ    ≡ 1/Γ(Σ^+→ pμ^+μ^-)dΓ(Σ^+→ pμ^+μ^-)/dM_μμ .
 
Its dependence on the dimuon invariant mass  M_μμ=√(q^2)  is depicted in Fig. <ref>. 
These distributions have  been normalized to their respective rate. 
It can be seen that the spectrum for solution 4, in both the relativistic and heavy-baryon cases, peaks at lower M_μμ than the other three.

  


The fraction of longitudinally polarized dimuons [and that normalized to the rate] can be written as 

    f_L^    = 2/3( F_0- F_2)    [ f̂_L^ ≡ f_L^/Γ(Σ^+→ p μ^+μ^-)]  ,

where explicit expressions for F_0 and F_2 in terms of the general parameters in Eq. (<ref>) can be found in the appendix of Ref. <cit.>. As discussed in that paper, the differential rate, Γ', is proportional to  F_0+ F_2/3  and is dominated by F_0 numerically. 
Yet, it may be possible to extract the f_L^ term from Eq. (<ref>) experimentally, as was done for  Λ_b→Λμ^+μ^-  by LHCb <cit.>. 
We illustrate the SM fourfold ambiguity for this quantity in Fig. <ref>.
The f_L^ distribution mostly mimics the dimuon spectrum. 
This is expected due to the numerical smallness of F_2
relative to F_0 <cit.> within the SM.








§ NEW PHYSICS CONTRIBUTIONS


To take into account the effects of new physics beyond the SM, we work with the effective Hamiltonian 

    H_ eff^    =  -∑_i 𝒞_i O_i+  H.c. ,

where the operators O_i are defined as 

    O_7^    = λ_t^G_ F^/√(2) e m_s^/4π^2 s_L^σ^κνd_R^  F_κν^ ,     O_7^'^    = λ_t^G_ F^/√(2) e m_s^/4π^2 s_R^σ^κνd_L^  F_κν^ , 
    
     O_9^ℓ    = λ_t^G_ F/√(2) e^2/4π^2 s_L^γ^ν d_L^ ℓγ_ν^ℓ ,     O_9^'^ℓ    = λ_t^G_ F^/√(2) e^2/4π^2 s_R^γ^ν d_R^ ℓγ_ν^ℓ , 
     O_10^ℓ    = λ_t^G_ F^/√(2) e^2/4π^2 s_L^γ^ν d_L^ ℓγ_ν^γ_5^ℓ ,     O_10^'^ℓ    = λ_t^G_ F^/√(2) e^2/4π^2 s_R^γ^ν d_R^ ℓγ_ν^γ_5^ℓ , 
    
     O_S^    = λ_t^G_ F^/√(2) e^2m_s^/4π^2 s_L^ d_R^ ℓℓ ,     O_S^'^    = λ_t^G_ F^/√(2) e^2m_s^/4π^2 s_R^d_L^ ℓℓ , 
    
     O_P^    = λ_t^G_ F^/√(2) e^2m_s^/4π^2 s_L^ d_R^ ℓγ_5^ℓ ,     O_P^'^    = λ_t^G_ F^/√(2) e^2m_s^/4π^2s_R^d_L^ ℓγ_5^ℓ ,
 
with  f_L,R^=P_L,R^ f  for fermion f and  λ_ q=V_ qd^V_ qs^*.  
For the Wilson coefficients 𝒞_i^ in Eq. (<ref>), we explicitly separate any SM contribution to them by writing  𝒞_i^=C_i^ SM+C_i^,  where "C_i" designates the part induced by NP.
Since we consider only NP that preserves CP symmetry, we suppose that the products  λ_t^C_i^,  which occur in H_ eff, are real numbers.
As seen below, the C_i enter the kaon observables of interest as either of  C_j±≡(C_j± C_j^'),  j=7,9,10,S,P,  and so we will adopt those combinations.


From the H.c. part of H_ eff in Eq. (<ref>), the NP contributions are

    ã    = λ_t^G_ F^/√(2) e^2m_s^/4π^2 q^2  c_σ^  C_7+^ ,    b̃    = λ_t^G_ F^/√(2) e^2m_s^/4π^2 q^2  c_σ^  C_7-^ , 
    c̃    = -λ_t^G_ F^/√(2) e^2/8π^2  C_9+^ ,    d̃    = λ_t^G_ F^/√(2) e^2/8π^2 (F-D) C_9-^ , 
    ẽ    = λ_t^G_ F^/√(2) e^2/8π^2  C_10+^ ,    f̃    = λ_t^G_ F^/√(2) e^2/8π^2 (D-F) C_10-^ , 
    g̃    = λ_t^G_ F^/√(2) e^2m_s^/8π^2 m_Σ^-m_p^/m_s^-m_d^  C_S+^ ,    j̃    = λ_t^G_ F^/√(2) e^2m_s^/8π^2 m_Σ^-m_p^/m_d^-m_s^  C_P+^ ,

and

    h̃    = λ_t^G_ F^/√(2) e^2 m_s^/8π^2 m_Σ^+m_p^/m_K^2-q^2 m_K^2/m_d^+m_s^ (D-F) C_S-^ , 
    k̃    = λ_t^G_ F^/√(2) e^2/8π^2 m_Σ^+m_p^/q^2-m_K^2 (D-F)( m_K^2 m_s^/m_d^+m_s^  C_P-^ + 2 C_10-^ m_ℓ^)  ,

where the relation  λ_t^*C_i^*=λ_t^C_i^  has been applied, which is valid for  Im(λ_tC_i)=0. 
In numerical work we set  e^2 = 4πα_ e^(0)  for the a, b, c, and d terms in Eq. (<ref>), which arise from photon-exchange LD effects, while  e^2 = 4πα_ e^(m_Z)  in the rest of Eq.  (<ref>) and in Eqs.  (<ref>)-(<ref>).

The complete numerical expression for B(Σ^+→ pμ^+μ^-) can be written as follows

    B(Σ^+→ pμ^+μ^-)_i^    = [ SM_i+ ∑_j n_ij^ (λ_t^C_j) +10^-2 ∑_jk q_jk^ (λ_t^ C_j)(λ_t^ C_k)] × 10^-8 ,

where the term SM_i refers to each of the four solutions in the relativistic (heavy baryon) formulation. 
The terms linear in the Wilson coefficients (WC) representing new physics arise from interference with the SM and are therefore different for each case. 
They are largest when they arise from the  SM long-distance contribution, and we display our numerical results  in Table <ref>.



The terms quadratic in the WC, C_j C_k, arise purely from NP and are thus the same for all four solutions. 
The factors of λ_t inf Eq. (<ref>), suggest that, with this normalization, the NP WC  must reach at least 10^3 to modify the SM rates at the percent level or higher.[Equivalently, only NP arising from ultraviolet models without a mixing angle suppression similar to λ_t can measurably affect this rate.] 
We display our numerical results for these terms in Table <ref>.[Ref. <cit.> gives a result that includes only the C_9+^2 and C_9-^2. 
Our numbers disagree with theirs by factors of two. 
Converting their result to our notation, they find coefficients 1.94 for C_9+^2 and 0.94 for C_9-^2 instead of our 0.97 and 2.06, respectively, from Table <ref>.]



We  turn our attention to the integrated forward-backward asymmetry as defined in Eq. (<ref>) in the presence of NP. The decay rate that normalizes this asymmetry also varies when there is new physics as per Eq. (<ref>), but we present results that include the NP terms only in the numerator of the asymmetry. 
The rationale for this choice is that this observable will only be measured after the decay rate is known. 
With this in mind we write for the terms that arise from interference with the LD SM contribution as

    Ã_ FB    = ∫ A_ FB^ dq^2/Γ(Σ^+→ pμ^+μ^-)
        = [ a_10+(λ_t^C_10+)+a_10-(λ_t^C_10-)+a_S+(λ_t^C_S+)+a_S-(λ_t^C_S-) ] × 10^-2 ,

where the coefficients a_i for the four relativistic and heavy-baryon solutions are tabulated in Table <ref>.[These numbers appear in <cit.> for one relativistic solution, and when converted  to our conventions they disagree with ours. That reference comments that this is due to the use of different form factors.]  



With NP coefficients at the 10^4 level, their direct contributions can be comparable to the SM. The additional terms in the forward-backward asymmetry, Eq. (<ref>), are of the form

    Ã_ FB    = ∑_ij q̃_ij^ (λ_tC_i)(λ_tC_j)× 10^-5,

and our numerical results for  q̃_ij^ are given in Table <ref>. 






§ KAON OBSERVABLES


The kaon modes originating from the quark level transition  s→ dμ^+μ^-,   are also dominated by long-distance contributions within the SM. 
This makes it very difficult to extract precise information on the SM short-distance parameters, or, indeed, new physics. Nevertheless, the corresponding kaon data can serve as useful constraints on NP which is large enough to measurably affect the  Σ^+→ pμ^+μ^-  rate. 
The SM SD contributions enter these modes via the Wilson coefficients, at the scale  μ=1 GeV,

    C_7^ SM    = -λ_u/2λ_t  C_7γ^ ,     
        C_9^ SM    = 2π/α_ e( y_7V^-λ_u/λ_t  z_7V^)  ,    
        C_10^ SM    = 2π/α_ e  y_7A^ ,
    
where  λ_q=V_qd^V^*_qs.  
Numerically we employ  z_7V^ = -0.046α_ e^,   y_7V^ = 0.73α_ e^,  and  y_7A^ = -0.68α_ e^, as before.[The error in these coefficients is sometimes quantified as y_7V^ = (0.73±0.04) α_ e  and  y_7A^=(-0.68± 0.03) α_ e^,  with the uncertainty range arising from taking  m̅_t(m_t)=(167±5) GeV  and  0.8≤μ/ GeV≤1.2 <cit.>.]




 §.§ Relevant kaon modes


We begin with the modes K_L,S→μ^+μ^-. For our numerical study, we employ the expressions for all contributions to these two rates that exist in the public SuperIso code <cit.>. In both cases, the long-distance contribution is dominated by a two-photon intermediate state. Its extraction relies on the measured K_L,S→γγ rates supplemented with hadronic models.  The long-distance contributions to  K_L,S→μ^+μ^- are therefore potentially contaminated by NP entering through C_7,7^'. This has been considered in the literature and we list the corresponding constraints obtained in <cit.> on Table <ref> . There are several proposals to improve these measurements <cit.>.



  §.§.§ K_L→μ^+ μ^-


This mode has a short-distance contribution proportional to Re C_10^ SM, but the rate is dominated by a long-distance contribution from a two-photon intermediate state. The absorptive part of this LD contribution is determined by the measured rate for K_L→γγ, and the difference between the measured rate and this absorptive contribution can be attributed to a combination of a model-dependent LD dispersive contribution, a SM SD contribution, and new physics. All these terms can be  parametrized as follows <cit.>

    B(K^0_L→μ^+ μ^- )    =  τ_L f_K^2 m_K^3 β_μ,K/ 16 π(|N_L^ LD+A_L^ SM+A_L^ NP|^2+β_μ,K^2|B_L|^2) , 
    
    N_L^ LD=   ±[0.54(77) - 3.95 i]× 10^-11 (GeV)^-2 , 
    
    A_L^ SM=   ( G_ Fα/√(2)π)2m_μ/m_K( Y_c/s_W^2 Reλ_c+Y(x_t)/s_W^2 Reλ_t) , 
    
    A_L^ NP=    -( G_ Fα/√(2)π) [2m_μ/m_K Re(λ_tC_10-) +  m_K/m_s +m_d m_s Re(λ_tC_P-)] , 
    
    B_L=   ( G_ Fα/√(2)π)  m_K/m_s +m_d m_s Im(λ_tC_S-)  .

Recall that we are assuming no CP violation in the new physics for our numerical study, implying  that we take B_L=0 for example. We have explicitly separated the SM short-distance contribution,  A_L^ SM <cit.>, so that C_10 refers to NP only. 



  §.§.§ K_S→μ^+ μ^-


This mode also has SD and LD contributions entering the branching ratio as <cit.>

    B(K^0_S→μ^+ μ^- )    =  τ_S f_K^2 m_K^3 β_μ,K/ 16 π(β_μ,K^2|N_S^ LD-A_S^ NP|^2+|B_S^ SM+B_S^ NP|^2) , 
    
    N_S^ LD=    (-2.65 + 1.14 i )× 10^-11 (GeV)^-2 , 
    
    A_S^ NP=   ( G_ Fα/√(2)π)  m_K/m_s +m_d m_s Re(λ_tC_S-) , 
    
    B_S^ SM=   ( G_ Fα/√(2)π)2m_μ/m_K(- Y_c/s_W^2λ_c-Y(x_t)/s_W^2λ_t) , 
    
    B_S^ NP=   ( G_ Fα/√(2)π)[2m_μ/m_K(λ_t C_10-) +  m_K/m_s +m_dm_s(λ_t  C_P-)]  .

Again, we have  explicitly written the SM SD contribution through C_10 as B_S^ SM. 
In the scenario with no CP violation in the NP, B_S^ NP=0.



  §.§.§ K_L→π^0ℓ^+ℓ^-


Within the SM these modes are predominantly CP-violating and receive multiple contributions. 
There are direct and through mixing CP-violating contributions as well as interference between them,  C_ dir^ℓ, C_ mix^ℓ and C_ int^ℓ respectively. 
Theoretical considerations suggest that the sign of the interference term is plus <cit.>. 
There is also a LD CP conserving contribution dominated by a two-photon intermediate state,[For our numerical studies we will use the expressions for these modes as coded for SuperIso and provided to us by F. Mahmoudi.] C_γγ^ℓ. 
Potentially there are NP contributions as well, which we include following <cit.>. Some enter C_ dir^ℓ and others we dub C_ NP^ℓ. 
A practical complication when using these modes to constrain NP is that two contributions are extracted from experiment. 
The CP conserving contribution is partially extracted from K_L→π^0γγ and as such is potentially contaminated by NP through C_7,7^'. 
We partially address this issue by using the constraints on C_7 and C_7^' as obtained from K→ππγ and K→γγ modes respectively <cit.>. 
The CP violating contribution through mixing relies on the measurement of K_S→π^0ℓ^+ℓ^- and NP affecting those modes could contaminate the extracted value of a_S. 
However, an a_S representing LD contributions is the same for both electron and muon modes so that the difference between the two extractions can be attributed to NP that breaks lepton flavor universality (LFU).

We follow Refs. <cit.> to write,

    B(K_L →π^0 ℓℓ̅) = ( C_ NP^ℓ+C_ dir^ℓ± C_ int^ℓ|a_S| + C_ mix^ℓ|a_S|^2 + C_γγ^ℓ)× 10^-12 .

As the expressions below imply, NP contributions may enter the rate through the term C_ NP via the coefficients C_P+ and C_S+ and also through the term C_ dir via the coefficients C_9+ and C_10+. 

To exhibit the dependence of the first three terms in Eq. (<ref>) on the possible NP, we follow Refs. <cit.> and write

    C_ NP^μ=    10^8  Im (λ_t C_P+)(0.09  Imλ_t +(0.0039   Im (λ_t C_P+)-0.02  Im(λ_t C_10+))) , 
    
         +     10^4  Re (λ_tC_S+)(-0.027(7)+0.0019× 10^4  Re(λ_t C_S+)) , 
    
        C_ dir^μ=   10^8(0.033( Im(λ_t C_10+)-4.27(19) Imλ_t)^2+0.014( Im(λ_t C_9+)+4.59(25) Imλ_t)^2) , 
    
        C_ int^μ=     10^4(1.37(8) Imλ_t+0.3 Im(λ_t C_9+)) , 
    
        C_ NP^e =   10^8  Im (λ_t C_P+)(0.0013 Imλ_t +(0.0076   Im (λ_tC_P+)-0.0003 Im(λ_tC_10+))) , 
    
         +     10^4  Re (λ_t C_S+)(-0.021(4)+0.0077× 10^4  Re(λ_t C_S+)) , 
    
        C_ dir^e=    0.06× 10^8(( Im(λ_t C_10+)-4.27(19) Imλ_t)^2+( Im(λ_t C_9+)+4.59(25) Imλ_t)^2) , 
    
        C_ int^e=    10^4( 5.84(35) Imλ_t+1.28 Im(λ_tC_9+) ) ,

where the terms linear in Re C_S+ arise from interference with the two-photon contribution. The explicit factors of 10^4 (10^8) appear from the practice of writing this result in terms of 10^4 λ_t. In addition, the last two terms in Eq. (<ref>) are given by

    C_ mix^μ    = 3.36 ± 0.20  ,     C_γγ^μ    = 5.2 ± 1.6  ,     
        C_ mix^e     = 14.5± 0.5  ,     C_γγ^e    ≈ 0  .

In the scenario we are pursuing here, where the NP is CP conserving, these results indicate that C_S+ is the only WC that can be constrained by these modes. 

However, it is also possible for NP to contaminate the extraction of the parameter a_S and in that way a constrain on C_9+ arises. The parameter a_S can be inferred from measurements of K_S→π^0ℓ^+ℓ^-. 
Those modes, in turn,  are LD dominated and parametrized  in χPT with the constants a_S, b_S which are then extracted from experiment <cit.>. Early extractions assuming LFU suggested the value  |a_S| = 1.20 ± 0.20 <cit.>, whereas more recent studies  suggest a larger error, |a_S|=1.3±3.2 <cit.>.  

A contamination from NP can only be extracted from these if it violates LFU.
The separate measurements of a_S in the muon and electron modes start from the χPT parametrization of the LD SM rates as <cit.>

    B(K_S→π^0e^+e^-)=   (0.01-0.55a_S-0.17b_S+43.76a_S^2+11.83a_S b_S+1.28b_S^2)× 10^-10 ,
     B(K_S→π^0μ^+μ^-)=   (0.07-3.96a_S-1.34b_S+86.90a_S^2+50.21a_S b_S+7.66b_S^2)× 10^-11 .

Existing measurements do not have enough information to determine both a_S and b_S so predictions rely on a vector meson dominance model leading to b_S/a_S=m_K^2/m_ρ^2 <cit.>.  
Using this form, the experimental collaborations have extracted a value for a_S from each of the two modes <cit.>,

    a_S  from  K_S→π^0 e^+e^-       1.06^+0.26_-0.21±0.07
    
                    a_S  from  K_S→π^0μ^+μ^-        1.54^+0.40_-0.32±0.06

With these two numbers one can constrain the combination C^μμ_9+-C^ee_9+,

    (a_S^μμ-a_S^ee) =    - √(2)  Re( V_tdV^*_ts(C^μμ_9+-C^ee_9+) )
    
        =    - √(2) |λ_t|( C^μμ_9+-C^ee_9+) = 0.5±0.5

A forward-backward asymmetry in these modes could also be used to constrain NP <cit.>, but there is no experimental information yet.



  §.§.§ K^+→π^+μ^+μ^-


Within the SM, this decay is also dominated by LD contributions that arise from an intermediate photon state. The LD contributions to these modes are parametrized by the low-energy constants a_+ and b_+ which are themselves extracted from the measurements, they are <cit.>,

    B_ LD^e=   (0.15-3.31a_+-0.90b_++60.51a_+^2+16.36a_+b_++1.77b_+^2)× 10^-8 ,
     B_ LD^μ=   (1.19-19.97a_+-6.56b_++120.16a_+^2+69.42a_+b_++10.59b_+^2)× 10^-9 .

The values of a_+ and b_+  extracted from experiment for the electron mode <cit.> are

    a_+=-0.578±0.016,   b_+=-0.779±0.0.066,

and for the muon mode  <cit.> 

    a_+=-0.575±0.013,   b_+=-0.722±0.0.043.

We add possible contributions from NP to the branching ratio resulting, for the muon case, in

    10^8  B   (K^+→π^+μ^+μ^-)
    = B_ LD^μ + 9.66    (λ_t C_7+^) +34   (λ_t C_9+^)+17.6 (λ_t C_7+^)(λ_t C_9+^)
    
    -   48.4  (λ_t C_10+^)(λ_t C_P+^)
    + 2.5  (λ_t C_7+^)^2+ 31 (λ_t C_9+^)^2+74.3  (λ_t C_10+^)^2
    
    +    9.46  (λ_t C_P+^)^2 +4.34  (λ_t C_S+^)^2.

This has been partially considered in  <cit.>, which provides an expression that includes only the term C_9+^2 with a numerical coefficient approximately twice as large as ours.[This factor can be traced to an incorrect √(2) in the form factor in their Eq. (4.16).]

A forward-backward asymmetry  can be defined for this mode in terms of the angle θ_Kμ between the charged kaon and the muon three momenta in the di-lepton center of mass frame <cit.>,

    A_FB,K^+=    N(cosθ_Kμ>0)- N(cosθ_Kμ<0)/ N(cosθ_Kμ>0)+ N(cosθ_Kμ<0).

In terms of the NP coefficients the asymmetry is  given by[Here we differ from <cit.> in that we normalise the forward-backward asymmetry to a fixed rate, the current experimental value B(K^+→π^+μ^+μ^-)=(9.17± 0.14)× 10^-8 <cit.>.]

    A_FB,K^+=   0.51  (λ_t C_S+^) + 0.844 (λ_tC_9+^)(λ_tC_S+^)+0.24  (λ_tC_7+^)(λ_tC_S+^)

The most recent measurements find the branching ratio to be  B(K^+→π^+μ^+μ^-)_ exp=(9.17± 0.08) ×10^-8 and  the forward-backward asymmetry A_FB,K^+=(0.0± 0.7)× 10^-2 <cit.>. 

Constraints on NP must rely on comparisons between the muon and electron modes and arise from LFUV. 
One possibility  is to take the difference between the branching ratio B(K^+→π^+μ^+μ^-) evaluated with  a_+,b_+ extracted from the muon mode, Eq. <ref>, and evaluated  with  a_+,b_+ from the electron mode, Eq. <ref>, 

    Δ B(K^+→π^+μ^+μ^-)=(-4±5)× 10^-9,

and to attribute this difference to the NP terms in Eq. <ref>.

A second possibility is to compare the parameter a_+ as extracted from muon and electron modes <cit.>,

    (a_+^μμ-a_+^ee) = - √(2) (V_tdV^*_ts (C^μμ_9+-C^ee_9+) ).

Using the current values of a_+^μμ,ee <cit.> this results in

    (a_+^μμ-a_+^ee)=0.003± 0.021.


 


§ ORGANIZING THE CONSTRAINTS


There are ten distinct Wilson coefficients to describe NP in modes with muons and ten more in modes with electrons if we allow LFUV. Fortunately, as Table <ref> indicates, many of the modes depend only on one or two WC, simplifying the numerical study. 

A direct constraint on C_7,7^' can be obtained in principle from radiative hyperon and kaon decays <cit.>. However, our method to estimate the long-distance contributions to hyperon decay involves using the a and b parameters obtained from experiment, entailing that these could be contaminated by any NP of this form. In principle, a more detailed study of all radiative hyperon modes could be used to isolate any NP here, but we will not pursue it in this paper. Instead, we rely on the constraints obtained from radiative kaon modes by <cit.>, as listed in Table <ref>.

Eq. (<ref>) suggests that NP scenarios with  C_7,7^'∼ 10^5, C_9,9^'∼ 10^4, C_S,S^'∼ 10^3-4 and/or C_10^('),P^(')∼ 10^5 would be necessary to affect Σ^+→ p μ^+μ^- at a level that is observable over the long-distance contributions. This is significantly larger than the bounds placed by kaon physics, so we will first impose the kaon constraints and then use Σ^+→ p μ^+μ^- to constrain the directions to which the kaons are blind. We will not delve with specific UV completions in this paper, but simply note that  WC of this order would be consistent with dimensional analysis in models without the CKM suppression λ_t that occurs in the SM and that was rather arbitrarily introduced by the notation of Eq. (<ref>).

Table <ref> summarizes the WC that can introduce NP into the different kaon observables discussed thus far.  



The observables in Table <ref> are constrained numerically by the 90% confidence-level (CL) bounds (ranges) <cit.> shown in Table <ref>, which also shows the corresponding limits for WC for the cases in which only one of them is constrained. K_L→μ^+μ^- constrains a combination  of C_10- and C_P-, which we dub C_KL,

    C_KL=     (C_10- +  m_K_0^2m_s/2m_μ(m_s +m_d)  C_P-)∼ C_10-+1.17  GeV C_P-.

Interestingly, this is the same combination that appears in the form factor k̃ in Σ^+→ p μ^+μ^-. However, the hyperon mode also depends on  C_10-  through  f̃.  The last two rows in  Table <ref> show bounds taken directly from Ref. <cit.>. 



After employing all the single parameter bounds from Table <ref>, we are left with the following constraints

    -1.2≤ 10^8 Δ B(K^+→π^+μ^+μ^-)≤ 0.4 
    
        |A_FB,K^+(K^+→π^+μ^+μ^-)|≤ 0.12 
    
        0.26 ≤ 10^8  B(Σ^+ → p μ^+μ^-)≤ 5.2


Because the constraint from K_L→μ^+μ^- is much stronger than what comes out of the Σ^+ decay, we present the results on C_10- and C_P- in terms of C_KL and its orthogonal counterpart C_Σ≡ C_10--0.85  GeVC_P-.

The result of imposing all the constraints simultaneously leads to allowed regions in C_10+-C_P+ and C_Σ-C_9- that differ by an order of magnitude. We thus display them separately in  Fig. <ref>. 
The C_10+ - C_P+ region is dominantly constrained by Δ B(K^+→π^+μ^+μ^-), whereas B(Σ^+ → p μ^+μ^-) restricts the parameters C_Σ and C_9-. For the latter, each of the four SM solutions leads to slightly different constraints as illustrated on the right panel of Fig. <ref>. The figure shows two-dimensional  projections of the full parameter space, but the numerical scan requires all parameters to satisfy all the constraints in Table <ref> and Eq. (<ref>). 

 

The most important NP contribution to B(Σ^+ → p μ^+μ^-) enters via C_9-, as suggested by the numerical entries in Table <ref>. The range of C_9- that is constrained by each of the four relativistic solutions is shown on the left panel of Figure <ref>. The right panel shows the correlation between the  integrated forward-backward asymmetry and the branching ratio for each of the four solutions. 



The resulting integrated forward backward asymmetry for Σ^+→ p μ^+μ^- across the parameter space that survives all the constraints, is also most sensitive to C_10- as can be inferred from Table <ref>, and we show it in the right panel of Figure <ref> for the four relativistic SM LD solutions. The left panel shows the correlation between the asymmetry and C_9-, where the variation with C_9- is due to the changing rate. Table <ref> suggests that the asymmetry would be equally sensitive to C_10+, but this is not observed in the results due to the much more restricted allowed range for this parameter.





§ SUMMARY AND CONCLUSIONS


We have revisited all aspects of the calculation of the rate and forward-backward asymmetry for the mode Σ^+→ p μ^+μ^-. 
This was motivated by the expected more precise measurement by LHCb as well as recent progress in related modes by BESIII. 

Within the SM, we corrected a long-standing issue with the SD contribution, but found that it is still negligible with respect to the LD contribution. For the latter, we updated existing experimental input which reduced the error in the predictions that is attributable to experimental error. The fourfold ambiguity in the prediction remains, but we proposed several ways to address this issue from the experimental side. With the new numbers, the estimate for hadronic error obtained by interpolating the leading order relativistic and heavy-baryon χPT results is also diminished, to the extent that the four separate solutions remain separate throughout the range. 

We then discussed possible constraints on new physics in a model independent manner through a general low-energy effective Hamiltonian for s→ d ℓ^+ℓ^- processes. We considered the constraints that can be placed on the different Wilson coefficients from existing measurements in rare kaon decay and emphasized the usefulness of the hyperon mode in probing the remaining blind directions. Aspects of this study exist in the literature but this is the first such comprehensive study. Ref. <cit.>, for example, has done a partial study of the effects of NP in the hyperon mode and the constraints from kaons. Their NP analysis of the hyperons does not contain all the possible WCs, missing in particular the interference between SM and NP in the rate which provides the most stringent constraint on C_9-. Within the kaon modes, we introduced a new observable to constrain NP based on the K^+→π^+ℓ^+ℓ^- rates. 



We thank Cong Geng for experimental information. 
The work of A.R. and G.V. was supported in part by an Australian Research Council Discovery Project.
We are grateful to F. Mahmoudi for her help with SuperIso, which included providing us with her private code for some of the kaon modes discussed in this paper.
The work of J.T. was supported in part by the National Key R&D Program of China under Contract No. 2023YFA1606000 and the National Natural Science Foundation of China (NSFC) under Contract No. 11935018. 
He would like to thank Hai-Bo Li and the Institute of High Energy Physics, Chinese Academy of Sciences, for kind hospitality and support during the preparation of this paper. 

 





§ ADDITIONAL NUMERICAL INPUTS


All particle masses, widths and lifetimes as well as values for physics constants and CKM parameters have been taken from <cit.>. Other input is given in Table <ref>






