A dispersive approach to the CP conserving K → πℓ+ℓ− radiative decays
摘要
We reconsider the constraints on the form factors W+(s) and WS(s), describing the radiative decay modes K+ → π+ℓ+ℓ− and KS → π0ℓ+ℓ−, associated with the general properties of analyticity and unitarity. Starting from the simple consideration of the asymptotic behaviours of the two combinations 2W+(s) − WS(s) and W+(s) + WS(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 → 3π 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 → 3π 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 KS → π+π−π0 amplitude can be determined from the value of W+(0) + WS(0). The possibility of fixing the sign of WS(0) using experimental data on both |W+|2 and |WS|2 is discussed.