Pion-nucleus scattering and the Pauli principle

A density expansion of the pion self-energy for pions in nuclear matter is reexamined. It is shown that a single hole-line expansion of the self-energy (i) is equivalent to using a strongly quenched $\pi N$ scattering amplitude in the medium, and (ii) results in an inconsistent treatment of the virtual pions necessarily present in a field-theoretic description of the problem. Exchange of intermediate pions gives rise to nucleon-nucleon, as well as pion-nucleon scattering diagrams that both contribute to the pion self-energy in an essential way. The nucleon-nucleon scattering proceeds, for instance, via a one-pion-exchange potential that is, however, highly nonstatic for energy transfers between nucleons close to the incident energy. Such interactions are singled out automatically for special treatment in a field-theory approach to the problem, and should not be introduced in an ad hoc manner as part of an empirical $\mathrm{NN}$ interaction in nuclear matter. We evaluate the coherent and charge exchange contributions to the pion-nucleus optical potential, proportional to the total density and the neutron-proton density difference, respectively. The Pauli principle is found to provide a small correction to the coherent part, both in the hole-line and density expansion formalisms. However, the charge exchange part of the potential is almost completely damped at low energies in the hole-line expansion, while the inclusion of backward-going graphs (random-phase-approximation correlations) restores it to its value based on free space $\pi N$ charge exchange amplitudes (i.e., no net Pauli effect).

NUCLEAR REACTIONS Compared density and hole line expansions for pion-nucleus optical potential; influence of Pauli principle on effective amplitude.