  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 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
single particle energies, trying to keep the ground state energy sufficiently
well reproduced. An alternative route, that has been developed along 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 single particle dynamics and structure. This is the basis of
the particle-vibration coupling model. In this paper we present a formal theory
of the particle-vibration coupling model based on the Green' s function method.
The theory extends to realistic effective forces the macroscopic
particle-vibration coupling 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 single particle self-energy and the
phonon propagator. Critical aspects of the particle-vibration coupling model
are analysed in general. Applications at the lowest order of the expansion are
presented and discussed.

Nuclear Theory