ππparameters fromKe4data and a rigorous representation forππamplitudes

Previous attempts to extract the $\pi \pi$ scattering length $a_0$ from $K_{e4\mathrm{}}$ data have been handicapped by the lack of any one-parameter model for ${\delta }_0^0$ which is valid over the entire energy range of the data. In this paper, we present a simple but reliable one-parameter model for the $\pi \pi S$ waves and $P$ wave. The model is based on rigorous representations for $\pi \pi$ amplitudes, and is valid from threshold to $M_{\pi \pi }=400\mathrm{}$ MeV or more. We apply the model to all available $K_{e4\mathrm{}}$ data, and extract the value of $a_0$ together with the corresponding values for several other low-energy $\pi \pi$ parameters. We find that the world average of the data yields $a_0=(0.26\mathrm{}\pm 0.08){\mu }^{-1\mathrm{}}$ ${\mathrm{\mu }}^{-1\mathrm{}}$, in agreement with the current-algebra prediction of Weinberg. As an appendix, we present in detail the rigorous one-parameter equations for the $S$ waves and the $P$ wave upon which our simple model is based. We make a precise prediction for the $I=2\mathrm{}$ $S$ wave between threshold and 900 MeV, and we predict the $P$ wave for given values of the $\rho$ mass and width.