Pion-Pion Scattering Including Inelastic Intermediate States

The inverse amplitude dispersion relations for pion-pion scattering including inelastic intermediate states are solved for given models of the inelastic cross section. These models are calculated by representing the four-pion intermediate state as the combination of a three-pion resonance and a single pion. Interesting results are obtained using a small and slowly varying inelastic cross section, since this leads to a resonant behavior of $R=\frac{{\sigma }^{\mathrm{tot}}}{{\sigma }^{\mathrm{el}}}$ when the real part of the $P$-wave phase shift passes through $\pi$ in the inelastic region. The original solution of the pure elastic inverse amplitude equations exhibits a single sharp resonance in the $P$ wave, and the inclusion of a resonant $R$ in the iteration scheme results in the appearance of a second $P$-wave resonance. When the pion-pion coupling constant $\lambda \cong -0.1\mathrm{}$ and the three-pion decay coupling constant $G^2\cong 1$, the positions and widths of the two di-pion resonances are $M_{\zeta }\cong 600$ MeV, ${\Gamma }_{\zeta }\cong 30$ MeV and $M_{\rho }\cong 800$ MeV, ${\Gamma }_{\rho }\cong 50$ MeV. This value of $\lambda$ gives $S$ waves in good agreement with independent analyses of $\pi -N$ scattering data. These results prove that, at least for pion-pion scattering, solutions exist possessing two resonances with the same quantum numbers. When a rapidly rising inelastic cross section is used corresponding to a large $G^2$, the original elastic solution exhibiting a single $P$-wave resonance is changed inappreciably due to the dominance of the "nearby" inverse amplitude left cut.