Effect of Dipion Resonances on the Pion-Nucleon Scattering Amplitude

The effect of the newly discovered dipion resonance at about 730 MeV and of a possible one at about 400 MeV on pion-nucleon scattering is evaluated. Partial-wave dispersion relations are solved via the $\frac{N}{D}$ decomposition by matrix-inversion methods. Two choices of "phase-space" factors are considered. The results, as compared with earlier calculations of a similar type, show marked improvement in $P_{33}$ and $P_{11}$ scattering lengths (with the Martin-Uretsky-type phase-space factor) but continue to disagree violently with experiment on the $S_{\frac{1}{2}}T=\frac{1}{2}$ scattering length.