Dispersive treatment of $K_S\to \gamma \gamma$ and $K_S\to \gamma {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$

We analyse the rare kaon decays $K_S\to \gamma \gamma$ and $K_S \to \gamma\ell^+\ell^-$ $(\mathrm{\ell }=eor\mu )$ in a dispersive framework in which the weak Hamiltonian carries momentum. Our analysis extends predictions from lowest order $SU(3{)}_L\times SU(3{)}_R$ chiral perturbation theory ($\chi$PT${}_3$) to fully account for effects from final-state interactions, and is free from ambiguities associated with extrapolating the kaon off-shell. Given input from $K_S\to \pi \pi$ and $\gamma {\gamma }^{(\ast )}\to \pi \pi$, we solve the once-subtracted dispersion relations numerically to predict the rates for $K_S\to \gamma \gamma$ and $K_S\to \gamma {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$. In the leptonic modes, we find sizeable corrections to the $\chi$PT${}_3$ predictions for the integrated rates.