A dispersive approach to the CP conserving $K\to \pi {\ell }^{+}{\ell }^{-}$ radiative decays

Abstract

We reconsider the constraints on the form factors $W_{+}(s)$ and $W_S(s)$, describing the radiative decay modes $K^{+}\to {\pi }^{+}{\ell }^{+}{\ell }^{-}$ and $K_S\to {\pi }^0{\ell }^{+}{\ell }^{-}$, associated with the general properties of analyticity and unitarity. Starting from the simple consideration of the asymptotic behaviours of the two combinations $2W_{+}(s)-W_S(s)$ and $W_{+}(s)+W_S(s)$, we derive a minimal pair of dispersive representations which involves only two free parameters. An important input for these representations consists of the $K\to 3\pi$ decay amplitudes, for which we use a set of solutions of the Khuri-Treiman equations obtained recently. These solutions provide an extrapolation from the physical $K\to 3\pi$ decay region up to the resonant $K\pi \to \pi \pi$ scattering regions. We show that the experimental energy dependence of $|W_{+}{|}^2$ can be well reproduced and that the sign of $W_{+}$ is unambiguously determined. We also show that the yet unknown $\Delta I=1/2$ part of the $K_S\to {\pi }^{+}{\pi }^{-}{\pi }^0$ amplitude can be determined from the value of $W_{+}(0)+W_S(0)$. The possibility of fixing the sign of $W_S(0)$ using experimental data on both $|W_{+}{|}^2$ and $|W_S{|}^2$ is discussed.

Comments: Version2: figure 13 was corrected