Determination of the S-wave $\pi \pi$ scattering lengths from a study of $K^{\pm} \to \pi^{\pm} \pi^{0} \pi^{0}$ decays

We report the results from a study of the full sample of $~6.031 x 10^{7} K^{\pm} \to \pi^{\pm} \pi^{0} \pi^{0}$ decays recorded by the NA48/2 experiment at the CERN SPS. As first observed in this experiment, the $\pi^{0} \pi^{0}$ invariant mass (M_00) distribution shows a cusp-like anomaly in the region around $M_{00} = 2m_{+}$, where m_{+} is the charged pion mass. This anomaly has been interpreted as an effect due mainly to the final state charge exchange scattering process $\pi^{+}\pi^{-} \to \pi^{0} \pi^{0}$ in $K^{\pm} \to \pi^{\pm} \pi^{+} \pi^{-}$ decay. Fits to the M_{00} distribution using two different theoretical models provide the presently most precise determination of $a_{0}-a_{2}$, the difference between the pi pi S-wave scattering lengths in the isospin I = 0 and I = 2 states. Higher-order pi pi rescattering terms, included in the two models, allow also an independent, though less precise, determination of a_2.