Improved estimates of rareKdecay matrix elements fromKℓ3decays

The estimation of rare $K$ decay matrix elements from $K_{\ell 3}$ experimental data is extended beyond LO in chiral perturbation theory. Isospin-breaking effects at next-to-leading order (and partially next-to-next-to-leading order) in the chiral perturbation theory expansion, as well as QED radiative corrections, are now accounted for. The analysis relies mainly on the cleanness of two specific ratios of form factors, for which the theoretical control is excellent. As a result, the uncertainties on the $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and $K_L\to {\pi }^0\nu \overline{\nu }$ matrix elements are reduced by a factor of about 7 and 4, respectively, and similarly for the direct $CP$-violating contributions to $K_L\to {\pi }^0{e}^{+}{e}^{-}$ and $K_L\to {\pi }^0{\mu }^{+}{\mu }^{-}$. They could be reduced even further with better experimental data for the $K_{\ell 3}$ slopes and the $K_{\ell 3}^{+}$ branching ratios. As a result, the nonparametric errors for $\mathcal{B}(K\to \pi \nu \overline{\nu })$ and for the direct $CP$-violating contributions to $\mathcal{B}(K_L\to {\pi }^0{\ell }^{+}{\ell }^{-})$ are now completely dominated by those on the short-distance physics.