On the amplitudes for the CP-conserving K ± ( K S ) → π ± ( π 0 ) ℓ + ℓ − rare decay modes

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 short-distance behaviour. Finally, perspectives for future improvements in the theoretical understanding of these amplitudes are discussed.