/ 06 02 02 3 v 4 8 D ec 2 00 6 K → 3 π Final State Interactions at NLO in CHPT and Cabibbo ’ s Proposal to Measure a 0 − a 2

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 Cabibbo’s proposal 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 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 that it is not smaller than 5 % if added quadratically and 7 % if 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 of two-pion phase space factors which are physical. We compare our results for the cusp effect with Cabibbo and Isidori’s ones. Present address: Department of Physics, University of Illinois, Urbana IL 61801, USA. Present address: Center for Theoretical Physics, Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge MA 02139, USA.