Corrections to the closure approximation for the pion-nucleon T-matrix inside the nucleus

We give a solution to the three-body model of the optical potential of Revai, and Tandy, Redish and Bollé, which is valid in the case when the projectile is very light. The resulting potential includes corrections to the projectile-nucleon t-matrix which arise because the target nucleon is in interaction with the rest of the nucleus. We study the pion-carbon optical potential at energies below the 3,3 resonance by calculating the elastic differential cross section and reaction cross sections to the inelastic channels. As shown in the work of Tandy, Redish and Bollé the reactive content of the optical potential consists of two inelastic channels corresponding to nucleon knockout and nucleon pick-up. We use the pick-up channel as a model of pion absorption with one-nucleon emission, by treating the nucleon as a bound state of a pion and a nucleon.