Pion-nucleus scattering using finite binding potentials

A nonperturbative treatment of finite binding potentials in pion-bound nucleon scattering is developed. Coupled differential equations in configuration space are derived from a model wave function broken into two parts. Coordinates from the source of binding to the pion and to the nucleon are used in the elastic channel, while pion-nucleon center-of-mass coordinates are used for the excited channels. The excited channel equations are solved in closed form using a separable nonlocal pion-nucleon interaction. When the solution is inserted into the elastic channel equation, a relativistically modified single-particle equation for the elastically scattered meson is derived, which features an optical potential with both local and nonlocal parts. This model is used in conjunction with first order multiple scattering theory to construct pion-nucleus scattering. In the present application of the model the assumed pion-nucleon interaction is taken to be $s$ wave with threshold $p$-wave behavior. The region around the 33 resonance is numerically studied for both the three-body model and for pions on ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$. A consistent, inclusive treatment of many effects familiar from earlier work is obtained. Results from the three-body calculation show that the peak cross section is shifted upwards by about the binding energy of the struck nucleon. However, nonlinear effects drive the peak downwards when the three-body model is incorporated in the pion-nucleus formalism, so that several bound nucleons simultaneously scatter the meson. Effects of this kind are not expected to be sensitive to the use of an artificial $l=0\mathrm{}\pi N$ interaction. Resonances caused by the binding potential are found to be important in both the three-body and the $\pi -{}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ calculations. This suggests significant corrections to an impulse approximation approach.