# On the amplitudes for the CP-conserving $K^\pm(K_S)\to\pi^\pm(\pi^0)\ell^+\ell^-$ rare decay modes

## TL;DR

This paper develops a detailed theoretical framework for predicting the amplitudes of rare kaon decay modes involving leptons, aiming to facilitate searches for lepton-flavour universality violations.

## Contribution

It introduces a two-loop form factor representation and a phenomenological model combining resonance contributions and QCD constraints for these decay amplitudes.

## Key findings

- Constructed a two-loop representation of form factors.
- Proposed a phenomenological model with resonances and operator-product expansion.
- Discussed future improvements in theoretical predictions.

## Abstract

The amplitudes for the rare decay modes $K^\pm\to\pi^\pm\ell^+\ell^-$ and $K_S\to\pi^0\ell^+\ell^-$ 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.