Pion-pion scattering and the $K \to 3\pi$ decay amplitudes

We revisit the recently proposed method for the determination of the pi-pi scattering length combination a_{0}-a_{2}, based on the study of the pi0 pi0 spectrum in K+ -> pi+ pi0 pi0. In view of a precision measurement, we discuss here the effects due to smaller absorptive contributions to the K+ -> pi+ pi0 pi0 and K+ -> pi+ pi+ pi- amplitudes. We outline a method of analysis that can lead to a precision determination of a_{0}-a_{2} and is based on very general properties of the S matrix. The discussion of final-state rescattering and cusp effects is also extended to the two K_L -> 3pi coupled channels. Thanks to the present work, the theoretical error on the a_{0}-a_{2} combination extracted from the K+ -> pi+ pi0 pi0 spectrum is reduced to about 5%. A further reduction requires the evaluation of the effects arising from radiative corrections.