Threshold pion production in pion - nucleus collisions: A simple estimate

The Goldhaber-Teller model generalized to spin-isospin vibrations is used to provide a simple estimate for the total cross section for threshold pion production in pion-nucleus collisions (${\pi }^{-}{A}_Z\to {\pi }^{-}{\pi }^{+}{A}_{Z-1\mathrm{}}$) in the case of nuclei with $N=Z$. Cross sections are calculated using the threshold approximant to the production amplitude for single nucleons consisting of pion pole plus contact terms alternately derived from the phenomenological Lagrangian theory and the current-commutator theory of multiple-pion production. The threshold approximant in the latter theory fits the experimental pion-production data on protons poorly and, in the case of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, the lightest target nucleus we consider, that theory predicts cross sections about a factor of 2 smaller than those similarly calculated with phenomenological Lagrangian input. On the other hand, for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, the predictions of the alternative threshold theories are in essential agreement. In the case of the current-commutator theory, the Goldhaber-Teller predictions for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ are consistent with those of the particle-hole model obtained earlier by Eisenberg after quadrupling his calculated values to compensate for an omitted factor of 2 in his production amplitude. While the cross sections for the ($\pi , 2\pi$) reaction in nuclei are still expected to be quite small, the prospect for their accessibility seems reasonably improved.