On the amplitudes for the CP-conserving K±(KS) → π±(π0)ℓ+ℓ− rare decay modes

Abstract

The amplitudes for the rare decay modes K ± → π ± ℓ + ℓ − and K S → π 0 ℓ + ℓ − are studied with the aim of obtaining predictions for them, such as to enable the possibility to search for violations of lepton-flavour universality in the kaon sector. The issue is first addressed from the perspective of the low-energy expansion, and a two-loop representation of the corresponding form factors is constructed, leaving as unknown quantities their values and slopes at vanishing momentum transfer. In a second step a phenomenological determination of the latter is proposed. It consists of the contribution of the resonant two-pion state in the P wave, and of the leading short-distance contribution determined by the operator-product expansion. The interpolation between the two energy regimes is described by an infinite tower of zero-width resonances matching the QCD shortdistance behaviour. Finally, perspectives for future improvements in the theoretical understanding of these amplitudes are discussed. Contents 1 Introduction 1 2 Theory overview 5 2.1 The structure of the form factors in three-flavour QCD 5 2.2 Properties of the form factors at high momentum transfer 8 3 Low-energy expansion of the weak form factors 12 3.1 The form factors at one loop 12 3.2 The form factors beyond one loop 16 4 Extraction of a +,S and b +,S from recent data 16 5 The two-loop representation of the form factors 20 5.1 Construction of the form factors to two loops 21 5.2 Comparing W +,S;2L (z) and W +,S;b1L (z) 23 6 W + (z) beyond the low-energy expansion 25 6.1 The contribution from the two-pion state 26 6.2 Intermediate states with higher thresholds 28 6.3 The contribution from the factorized Q 7V operator 33 6.4 Evaluations of a + and b + 34 7 Summary, conclusions, outlook 35 7.1 Extracting a +,S and b +,S from recent data 35 7.2 The low-energy expansion of the form factors to two loops 35 7.3 Contribution from the two-pion state 35 7.4 Matching with the short-distance regime 36 7.5 Conclusion 36 A Numerical values 37 B The computation of ψ +,S (s) 37

Anatomy of kaon decays and prospects for lepton flavour universality violation

“…The parameters of the form factor V + (z) are individually fitted to the two available data sets. The data sets are in agreement for most values of z except for those around z = 0.3 [60]. However, a rescaling of the errors in that region by a factor of about 2.5 leads to an agreement between the two.…”

“…This can then be used for the SM computations of the corresponding branching fractions [50,58]. The assumption of a SM-like pattern while estimating the coefficients a + and b + is reasonable on account of dominant long-distance effects [59][60][61]. Thus, any information regarding New Physics contributions due to shortdistance physics is hidden and not immediately apparent by noting the individual values of the branching for each channel.…”

Anatomy of kaon decays and prospects for lepton flavour universality violation

“…The parameters of the form factor V + (z) are individually fitted to the two available data sets. The data sets are in agreement for most values of z except for those around z = 0.3 [42]. However, a rescaling of the errors in that region by a factor of about 2.5 leads to an agreement between the two.…”

A measurement of the K+ → π+μ+μ− decay

“…The K πee decay was first observed at the CERN PS by the Geneva-Saclay collaboration in 1975 [12], and subsequently measured by the E777 [13], E865 [14] and NA48/2 [15] experiments. A summary of form factor measurements can be found in [16]. Improved measurements of the K πµµ model-independent branching fraction and form factor parameters, based on the dataset collected in 2017-2018 by the NA62 experiment at the CERN SPS, are presented in the following.…”