K → 3 π Final State Interactions at NLO in CHPT and Cabibbo ’ s Proposal to Measure a 0 − a

We present the analytical results for the K → 3π final state interactions at next-to-leading order (NLO) in CHPT. We also study the recent Cabibbo’s proposal to measure the ππ scattering lenghts combination a0 − a2 from the cusp effect in the π0π0 energy spectrum at threshold for K+ → π+π0π0 and KL → π0π0π0, and give the relevant formulas to describe it at NLO. For that, we use the NLO CHPT expression to fit the real part of K → 3π to data while the ππ scattering lenghts are treated non-perturbatively. Using them, we make a quantitative estimate of the theoretical uncertaintity of the a0−a2 determination at NLO in our approach and obtain that it is not smaller than 5 % if added quadratically and 7 % if linearly for K+ → π+π0π0. One gets similar theoretical uncertainties if the neutralKL → π0π0π0 decays data below threshold are used instead. For this decay, there are very large theoretical uncertainties above threshold due to cancellations and these data cannot be used to get the scattering lenghts. All the numbers we present are in the isospin limit. We compare our results for the cusp effect with Cabibbo and Isidori’s results and discuss the differences.