Dispersive treatment of $K_S\to {\hat{I}}^3{\hat{I}}^3$ and $K_S\to {\hat{I}}^3{\ell }^{+}{\ell }^{-}$

Abstract

We analyse the rare kaon decays $K_S\to {\hat{I}}^3{\hat{I}}^3$ and $K_S\to {\hat{I}}^3{\ell }^{+}{\ell }^{-}$ $(\ell = e \mbox{ or } μ)$ 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 ($\ddot{I}‡$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 $K_S\to \ddot{I}€\ddot{I}€$ and $γγ^{(*)}\toππ$, we solve the once-subtracted dispersion relations numerically to predict the rates for $K_S\to {\hat{I}}^3{\hat{I}}^3$ and $K_S\to {\hat{I}}^3{\ell }^{+}{\ell }^{-}$. In the leptonic modes, we find sizeable corrections to the $\ddot{I}‡$PT${}_3$ predictions for the integrated rates.

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