Measuring the a0-a2 pion scattering lengths through K to 3 pi decays

Abstract

We discuss the recent Cabibbo's proposal to measure the pion-pion scattering lengths combination a0-a2 from the cusp effect in the pi0-pi0 energy spectrum at threshold for K+ to (pi0 pi0 pi+) and KL to (pi0 pi0 pi0). We estimate the theoretical uncertainty of the a0-a2 determination at NLO in our approach and obtain that it is not smaller than 5% for K+ to (pi0 pi0 pi+). One gets similar theoretical uncertainties if the neutral KL to (pi0 pi0 pi0) 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.