Pion Production and the Second Pion-Nucleon Resonance

Abstract

A model for the reaction $\ensuremath{\pi}+N\ensuremath{\rightarrow}2\ensuremath{\pi}+N$ 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 $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths. These parameters are chosen from a fit to the total cross section for ${\ensuremath{\pi}}^{\ensuremath{-}}+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}+{\ensuremath{\pi}}^{+}+n$. 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., ${\ensuremath{\pi}}^{+}+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{+}+n$, etc.) in the energy range from threshold to \ensuremath{\approx}500 Mev in good agreement with experiments. Angular distributions are predicted. These are in qualitative agreement with the ${\ensuremath{\pi}}^{+}$ angular distribution for ${\ensuremath{\pi}}^{\ensuremath{-}}+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}+{\ensuremath{\pi}}^{+}+n$. From these data it is suggested that the $S$-wave $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ 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 $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths found are ${a}_{0}=\ensuremath{-}0.290{\ensuremath{\mu}}^{\ensuremath{-}1}$, ${a}_{1}=0.122{\ensuremath{\mu}}^{\ensuremath{-}1}$, and $\frac{{a}_{2}}{{a}_{0}}=\frac{2}{5}$ by hypothesis.