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 $K\to3\pi$ 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.