Pion Production in Pion-Nucleon Scattering

The cross section for the production of one additional pion in a pion-nucleon scattering is calculated using the Chew-Low theory of $P$-wave pion-nucleon scattering. The transition matrix element for the scattering of one meson into two mesons is defined in terms of exact eigenstates of the total Hamiltonian and an approximate expression is derived which expresses the one- to two-meson matrix element as a product of an elastic scattering matrix element and a meson-emission matrix element. Experimental elastic scattering phase shifts are used in calculating the two-meson cross section. The results of this calculation are used to estimate the effect of two-meson states on elastic scattering. The contribution to the effective range for elastic pion-nucleon scattering is small. Cross sections are also obtained for all possible two-meson charge states from either a ${\pi }^{+}$ or a ${\pi }^{-}$ meson incident on a proton. Comparison is made with experiment and the recent theoretical work of Barshay.