K→3π final state interactions at NLO in CHPT and Cabibbo’s proposal to measure a0-a2

Abstract

We present the analytical results for the K→3π final state interaction phases at next-to-leading order (NLO) in CHPT. We also study the recent proposal of Cabibbo to measure the ππ scattering lengths combination a0-a2 from the cusp effect in the π0π0 energy spectrum at threshold for K+→π0π0π+ and KL→π0π0π0, and we give the relevant formulas to describe it at NLO. We estimate the theoretical uncertainty of the a0-a2 determination at NLO in our approach and obtain the result that it is not smaller than 5% if added quadratically and 7% if added linearly, for K+→π0π0π+. One gets similar theoretical uncertainties if the neutral KL→π0π0π0 decay data below threshold are used instead. For this decay, there are very large theoretical uncertainties above threshold due to cancellations, and data above threshold cannot be used to get the scattering lengths. We do not include isospin corrections apart from two-pion phase space factors which are physical. We compare our results for the cusp effect with those due to Cabibbo and Isidori.