Pion-Pion Scattering andK±→3πDecay

The effects of final state pion-pion interactions on the spectrum of $K^{\pm }\to 3\pi$ decay is studied by dispersion relation methods. In the approximations adopted we are led to a set of linear integral equations for the amplitudes of the $K^{\pm }\to 3\pi$ decay. The kernels in these equations depend on the pion-pion $S$-wave scattering amplitudes. An approximate solution for these equations is obtained by iteration and the departures from a purely statistical spectrum for the decay are related to pion-pion $S$-wave scattering. The latter in turn is assumed to be well represented with a scattering length structure. The $K^{\pm }\to 3\pi$ spectrum then is parametrized by two quantities, the $T=0\mathrm{}$ and $T=2\mathrm{}$ pion-pion $S$-wave scattering lengths, $a_0$ and $a_2$. Such experimental results as presently exist indicate that $a_2-a_0$ is positive and that roughly $a_2-a_0\approx 0.7$, in units of the pion Compton wavelength.