Local field corrections inπ-nucleus scattering

Abstract

The local field correction to the $\ensuremath{\pi}$-nucleus optical potential is calculated taking recoil into account. The correction arises from the triple scattering matrix element $〈0;\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|{\ensuremath{\tau}}_{1}G{\ensuremath{\tau}}_{2}G{\ensuremath{\tau}}_{1}|0;{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}}〉$, where ${\ensuremath{\tau}}_{i}$ are $\ensuremath{\pi}N$ amplitudes. The recoil energy of nucleon 1 affects the pion Green's function $G$ and the intermediate scattering amplitude ${\ensuremath{\tau}}_{2}$. The recoil of nucleon 2 affects only ${\ensuremath{\tau}}_{2}$ and has not been included to save computational time. Effects of antisymmetry of the two nucleons, spin flip, and charge exchange have been included, but not those of pair correlation and Pauli blocking. Inclusion of recoil is found to reduce the local field correction by a factor of 7 from the result of fixed scatterer calculations. Nevertheless, the local field correction remains sizable. In the $\ensuremath{\pi}$-$^{16}\mathrm{O}$ case it represents a correction $\frac{2}{3}$ as large as the first order optical potential near the $\ensuremath{\pi}N$ resonant energy. Nucleon 1 typically recoils through \ifmmode\times\else\texttimes\fi{} 1 fm while in terms of the internucleon separation ${r}_{12}$ the effect is very long ranged. Thus the $P$-wave nature of the $\ensuremath{\pi}N$ interaction which enhances the reflection of the pion from nucleon 2 back to nucleon 1 is the main reason why the local field correction is so important in $\ensuremath{\pi}$-nucleus scattering. The imaginary part of the local field correction part of the optical potential is analyzed and its reactive content is shown to arise from two competing mechanisms. The first, and dominant, mechanism is a reduction of the single nucleon knockout contribution that is overestimated in the first order optical potential while the second mechanism is the two-nucleon knockout. The analysis suggests very slow convergence of multiple scattering expansions near the $\ensuremath{\pi}N$ resonance energy.NUCLEAR REACTIONS Pion optical potential, local field corrections, reactive content of optical potential.