Theory of Low-Energy Pion-Pion Scattering. II

We have extended the previously published pion-pion calculations by solving numerically the coupled $s$-and $p$-wave equations based on a method of the inverse amplitude, to see if the lower partial waves near threshold converge to definite values. Elastic unitarity is still used for the right-hand cut. Numerical calculations show that the $s$-and $p$-wave scattering lengths are very much the same as those obtained before by using only the left-hand singularities chosen to have the correct functional forms near the branch point; however, the position and the width of the $p$-wave resonance energy are substantially improved. Solutions exhibit results as a function of $\lambda$ similar to those found previously; a $p$-wave attraction corresponds to a strong $s$-wave attraction. Thus we find that the procedure of constructing the amplitude developed in the earlier paper is convincing.