Pion-pion scattering and the K → 3π decay amplitudes

We revisit the recently proposed method for the determination of the ππ scattering length combination a0 − a2, based on the study of the π0π0 spectrum in K+ → π+π0π0 . In view of a precision measurement, we discuss here the effects due to smaller absorptive contributions to the K+ → π+π0π0 and K+ → π+π+π− amplitudes. We outline a method of analysis that can lead to a precision determination of a0 − a2 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 KL → 3π coupled channels. Thanks to the present work, the theoretical error on the a0 − a2 combination extracted from the K+ → π+π0π0 spectrum is reduced to about 5%. A further reduction requires the evaluation of the effects arising from radiative corrections.