Dyson's equations for nucleon-pion-delta interactions: Toward a self-consistent solution in nuclear matter

Abstract

A formalism is developed for the self-consistent solution of Dyson's equations for the pion and the delta in cold nuclear matter. The nucleon-delta and nucleon-pion “loop integrals” are evaluated and are inserted into the pion and delta-particle propagators, respectively. The resulting pion propagator displays two distinct families of “spikes” which are dramatically different from the free-space behavior. In particular, there are significant high-momentum components of the pion which give large contributions to the nucleon-pion loop integral. Such contributions lead to severe spikes in the iterated delta propagator. Studies are made too of the sensitivity of the results of the cutoff range of the “form factor” used to evaluate the loop integrals, to the starting solution, and to the density. Also, for the densities studies here, the pion propagator exhibits pure delta-function singularities, which occur over a small range of energies just above the mass of the pion. However, it is shown that such delta functions have a negligible effect on the main calculation. Finally, this paper gives a comprehensice treatment of the main methods used to solve the nuclear matter Dyson's equations.