Observation of a cusp-like structure in the pi0 pi0 invariant mass distribution from K+ --->pi+- pi0 pi0 decay and determination of the pi pi scattering lengths.

We report the results from a study of a partial sample of ∼2.3×107 K±→π ±π0π0 decays recorded by the NA48/2 experiment at the CERN SPS, showing an anomaly in the π0π0 invariant mass (M00) distribution in the region around M00=2m+, where m+ is the charged pion mass. This anomaly, never observed in previous experiments, can be interpreted as an effect due mainly to the final state charge exchange scattering process π+π -→π0π0 in K±→ π±π+π- decay [N. Cabibbo, Phys. Rev. Lett. 93 (2004) 121801]. It provides a precise determination of a 0-a2, the difference between the ππ scattering lengths in the isospin I=0 and I=2 states. A best fit to a rescattering model [N. Cabibbo, G. Isidori, JHEP 0503 (2005) 21] corrected for isospin symmetry breaking gives (a0-a2)m+=0.268±0. 010(stat)±0.004(syst), with additional external uncertainties of ±0.013 from branching ratio and theoretical uncertainties. If the correlation between a0 and a2 predicted by chiral symmetry is taken into account, this result becomes (a0-a2)m+=0. 264±0.006(stat)±0.004(syst)±0.013(ext).