Fairly Low-Energy Pion-Pion Scattering. II

We have extended the pion-pion calculation of Smith and Uretsky by including the third-order perturbation theoretic terms for the discontinuity across the left-hand cut in the complex energy plane. The right-hand cut is still given by elastic unitarity. Numerical calculations for the $S$- and $P$-wave amplitudes show that the $S$ waves are not much different than they were in the second-order calculation. The $P$-wave amplitude is substantially modified, and the trends are such as to make it plausible that a fourth-order calculation could reproduce the $\rho$ resonance. It was also interesting to find that the $P$-wave interaction can be strongly attractive (to this order) only if the $S$-wave interactions are repulsive.