Predictions for the rare kaon decays $K_{S,L} \to \pi^0 \ell^+ \ell^-$ from QCD in the limit of a large number of colours

Abstract

The long-distance and non-local parts of the form factors describing the single-photon mediated $K_{S,L}\to\pi^0\gamma^*\to\pi^0\ell^+\ell^-$ ($\ell = e, \mu$) transitions in the standard model are addressed in QCD in the limit where the number $N_c$ of colours becomes infinite. It is shown that this provides a suitable theoretical framework to study these decay modes and that it allows to predict the decay rates for $K_{S}\to\pi^0\ell^+\ell^-$. It also unambiguously predicts that the interference between the direct and indirect CP-violating contributions to the decay rate for $K_L\to\pi^0 \ell^+ \ell^-$ is constructive.