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

The long-distance and non-local parts of the form factors describing the single-photon mediated $K_{S,L}\to {\pi }^0{\gamma }^{\ast }\to {\pi }^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ ($\mathrm{\ell }=e,\mu$) 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 $K_S\to {\pi }^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$. It also unambiguously predicts that the interference between the direct and indirect CP-violating contributions to the decay rate for $K_L\to {\pi }^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ is constructive.