A dispersive approach to the CP conserving K → πℓ+ℓ− radiative decays

A bstract We reconsider the constraints on the form factors W + ( s ) and W S ( s ), describing the radiative decay modes K + → π + ℓ + ℓ − and K S → π 0 ℓ + ℓ − , associated with the general properties of analyticity and unitarity. Starting from the simple consideration of the asymptotic behaviours of the two combinations 2 W + ( 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 → 3 π 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 → 3 π decay region up to the resonant Kπ → ππ 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 ∆ I = 1 / 2 part of the K S → π + π − π 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.