Pion Production and the Second Pion-Nucleon Resonance

A model for the reaction $\pi +N\to 2\pi +N$ at low energies, which includes pion-pion interaction and final-state interactions in the (3,3) state, is discussed. The theory involves 2 parameters which are related to the $S$-wave and $P$-wave $\pi -\pi$ scattering lengths. These parameters are chosen from a fit to the total cross section for ${\pi }^{-}+p\to {\pi }^{-}+{\pi }^{+}+n$. Meson production is predicted to be primarily in the $T=\frac{1}{2}$ state. Predictions are made for the total cross sections of the various channels (e.g., ${\pi }^{+}+p\to {\pi }^{+}+{\pi }^{+}+n$, etc.) in the energy range from threshold to ≈500 Mev in good agreement with experiments. Angular distributions are predicted. These are in qualitative agreement with the ${\pi }^{+}$ angular distribution for ${\pi }^{-}+p\to {\pi }^{-}+{\pi }^{+}+n$. From these data it is suggested that the $S$-wave $\pi -\pi$ scattering length has opposite sign to the $P$-wave scattering length. A conjecture concerning rapidly rising inelastic cross sections in a single partial wave is made to connect the large $T=\frac{1}{2}$, $D_{\frac{3}{2}}$ production cross sections with the $T=\frac{1}{2}$, $D_{\frac{3}{2}}$ pion-nucleon resonance. The $\pi -\pi$ scattering lengths found are $a_0=-0.290\mathrm{}{\mu }^{-1\mathrm{}}$, $a_1=0.122\mathrm{}{\mu }^{-1\mathrm{}}$, and $\frac{a_2}{a_0}=\frac{2}{5}$ by hypothesis.