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Dispersive treatment of $K_S\toγγ$ and $K_S\toγ\ell^+\ell^-$

We analyse the rare kaon decays $K_S \to γγ$ and $K_S \to γ\ell^+\ell^-$ $(\ell = e \mbox{ or } μ)$ 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 ($χ$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 $K_S \to ππ$ and $γγ^{(*)}\toππ$, we solve the once-subtracted dispersion relations numerically to predict the rates for $K_S \to γγ$ and $K_S \to γ\ell^+\ell^-$. In the leptonic modes, we find sizeable corrections to the $χ$PT$_3$ predictions for the integrated rates.