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

Abstract

We analyse the rare kaon decays $K_S\to \gamma \gamma$ and $K_S\to \gamma {\ell }^{+}{\ell }^{-}$ $(\ell =e\text{mbox}or\mu )$ 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 $K_S\to \gamma {\ell }^{+}{\ell }^{-}$. In the leptonic modes, we find sizeable corrections to the $\chi$PT${}_3$ predictions for the integrated rates.

Comments: 12 pages, 9 figures. v2: Improved assessment of systematic uncertainties, some typos corrected, matches published version