Low density expansion of the nucleon-nucleus optical-model potential

The optical-model potential for nucleon-nucleus scattering is studied within the framework of the Green function approach to the many-body problem. The optical potential is identified with the self-energy, for which an expansion in terms of irreducible graphs exists. We propose to group the diagrams of this expansion according to the number of independent hole lines and to sum the graphs within each class. This procedure essentially amounts to an expansion in powers of the density and is closely related to the Bethe-Brueckner expansion for the binding energy of nuclear matter. We show that the same convergence parameter appears in both expansions. The one-and two-hole line contributions are studied in detail, numerical estimates are provided and compared with experiment. At low energy, our expansion can be related to the calculation of the optical potential within the framework of nuclear reactions (e.g., using doorway states). At high energy one is led in a natural way to the expressions derived from multiple scattering theory. Thus the hole-line expansion ties together the low and high energy domains of the optical potential. Hole line expansions for the momentum distribution and the total energy are derived from the expansion of the self-energy. The self-consistency requirement is discussed. The present study is restricted to nuclear matter but most results apply to finite nuclei as well.