Pion-Pion Scattering. III

Continuing a program discussed in two earlier papers, we have calculated the left-hand partial-wave discontinuities in fourth-order perturbation theory. The $s$-wave and $p$-wave dispersion relations are solved by standard $\frac{N}{D}$ techniques. It is found that the relative contributions arising from the double-spectral terms and the inelastic single-spectral functions are negligible. As a perturbation theory the procedure appears to be sensible for coupling constants $\left|\lambda \right|$ less than about 0.1. High-energy $p$-wave resonances can be made to occur for $\left|\lambda \right|$ of about 0.2, but with very large widths. It is suggested that the physical pion-pion system may not be describable in terms of a one-parameter theory, and this notion is given experimental meaning.