Low-Energy Pion-Pion Scattering

This calculation starts with the partial-wave dispersion relations for pion-pion scattering as deduced from the Mandelstam representation for the scattering amplitude. The contribution of the "left-hand cut" was calculated from $\lambda {\phi }^4$ perturbation theory (to second order in $\lambda$). The $s$-wave and $p$-wave amplitudes were obtained numerically and found to agree closely with those obtained by Chew, Mandelstam, and Noyes who proceeded from a different viewpoint.