Angular Correlations inKe4Decays and Determination of Low-Energyπ−πPhase Shifts

The study of correlations in $K_{e4\mathrm{}}$ decays can give unique information on low-energy $\pi -\pi$ scattering. To this end we introduce a particularly simple set of correlations. We show that the measurement of these correlations at any fixed $\pi -\pi$ c.m. energy allows one to make a model-independent determination of the difference ${\delta }_0-{\delta }_1$ between the $S$- and $P$-wave $\pi -\pi$ phase shifts at that energy. Information about the average value of ${\delta }_0-{\delta }_1$ can be obtained from a measurement of the same correlations averaged over the energy spectrum. Measurement of the average correlations is particularly suited to the testing of any model of low-energy $\pi -\pi$ scattering. We discuss in particular two such models: (a) the Chew-Mandelstam effective-range description of $S$-wave scattering and (b) the Brown-Faier $\sigma$-resonance model for the $S$ wave. If the Chew-Mandelstam description is adequate, the suggested measurements should yield a value for the $S$-wave scattering length in the $I=0\mathrm{}$ state. If the $\sigma$-resonance model is correct, these measurements should yield a value for the mass of the resonance.