Pion-Pion Interactions in the States T=0 and T=1

New dispersion relations are derived for the $s$-wave pion-nucleon scattering amplitudes. These relations are specifically chosen to facilitate the task of separating the two-pion exchange term from the other effects contributing to low-energy pion-nucleon scattering. In these equations the contribution from the unknown short-range terms is markedly suppressed, a greater emphasis is placed on the experimentally better established very low energy pion-nucleon data, in particular on the scattering lengths, and in the contribution of the two-pion exchange term the lower energies are more heavily stressed. The terms due to the two-pion exchange, which we isolate, are very clearly recognized by their characteristic energy dependences.The two-pion exchange terms are analyzed in terms of the interaction between the two pions. The values obtained for these terms are made to yield information about the phase shifts for pion-pion scattering in the $T=1$ and $T=0$ states.In the $T=1$ state the data are well fitted with a narrow $p$-wave resonance in the pion-pion system at 750 MeV. Taking the results of electron-nucleon scattering experiments in conjunction with the present data, the half-width of this resonance is found to lie between 40 and 60 MeV, in agreement with the experimental data for the observed $\ensuremath{\rho}$ meson.In the $T=0$ state the data are fitted with a two-parameter form for the $s$-wave pion-pion phase shift. Both of these parameters are not simultaneously determined by the present data. As an added restriction on the phase shift it is required to lead to agreement with the experimental results for the process $p+d\ensuremath{\rightarrow}{\mathrm{He}}^{3}+2\ensuremath{\pi}$. This singles out a $T=0$ $s$-wave phase shift with a scattering length of ${a}_{0}\ensuremath{\approx}1.6$ (units $\ensuremath{\hbar}=\ensuremath{\mu}=c=1$).