Dispersive description of the $K\to \pi {\ell }^{+}{\ell }^{-}$ radiative amplitudes

Abstract

We propose a description of the $K^{+}$, $K_S$ radiative decay form factors $W_{+}$, $W_S$ based on general properties of analyticity and unitarity. Starting from the simple consideration of the asymptotic behaviour of the two combinations $2W_{+}-W_S$ and $W_{+}+W_S$ we derive a dispersive representation involving only two parameters. Using the rich experimental information on the $K\to 3\pi$ amplitudes, extended beyond the low energy region using the Khuri-Treiman formalism, we show that the sign of the $W_{+}$ form factor is unambiguously determined and its energy dependence can be well reproduced. We also show that the yet unknown $\Delta I=1/2$ part of the $K_S\to {\pi }^{+}{\pi }^{-}{\pi }^0$ can be determined from the value of $W_{+}(0)+W_S(0)$. The possibility of fixing the sign of $W_S$ from experiment is discussed.

Comments: Talk given at the 13th International Conference on Kaon Physics (KAON2025), Mainz, 8-12 September 2025