Low-Energy Pion-Pion Scattering. II

A self-consistent calculation of low-energy $\pi -\pi$ scattering is made in which, in addition to the usual $P$-wave resonance, an $I=0\mathrm{}$, $D$-wave resonance is also retained. The only free parameter is the pion mass. The resulting resonances have masses of 685 and 892 MeV, respectively, and the half-width is about 160 MeV in each case. The procedure consists of combining the Chew-Mandelstam and generalized Ball-Wong techniques with self-consistency. An outline is also given of a possible generalization of such a procedure to an arbitrary $S$-matrix process.