Beyond the mean field in the particle-vibration coupling scheme

The energy density functional theory is one of the most used methods developed in nuclear structure. It is based on the assumption that the energy of the ground state is a functional only of the density profile. The method is extremely successful within the effective force approach, noticeably the Skyrme or Gogny forces, in reproducing the nuclear binding energies and other bulk properties along the whole mass table. Although the density functional is in this case represented formally as the Hartree–Fock mean field of an effective force, the corresponding single particle (s.p.) states in general do not reproduce the phenomenology particularly well. To overcome this difficulty, a strategy has been developed where the effective force is adjusted to reproduce directly the s.p. energies, trying to keep the ground state energy sufficiently well reproduced. An alternative route, that has been developed over several years, for solving this problem is to introduce the mean field fluctuations, as represented by the collective vibrations of the nuclear system, and their influence on the s.p. dynamics and structure. This is the basis of the particle–vibration coupling (PVC) model. In this paper we present a formal theory of the PVC model based on Green's function method. The theory extends to realistic effective forces the macroscopic PVC models and the (microscopic) nuclear field theory. It is formalized within the functional derivative approach to many-body theory. An expansion in diagrams is devised for the s.p. self-energy and the phonon propagator. Critical aspects of the PVC model are analysed in general. Applications at the lowest order of the expansion are presented and discussed.