  Self-consistent Green's function theory has recently been extended to the
basic formalism needed to account for three-body interactions [A. Carbone, A.
Cipollone, C. Barbieri, A. Rios, and A. Polls, (Phys. Rev. C 88, 054326
(2013))]. The contribution of three-nucleon forces has so far been included in
ab initio calculations on nuclear matter and finite nuclei only as averaged
two-nucleon forces. We derive the working equations for all possible two- and
three-nucleon terms that enter the expansion of the self-energy up to the third
order, thus including the interaction-irreducible (i.e., not averaged) diagrams
with three-nucleon forces that have been previously neglected. We employ the
algebraic diagrammatic construction up to the third order as an organization
scheme for generating a non perturbative self-energy, in which ring
(particle-hole) and ladder (particle-particle) diagrams are resummed to all
orders. We derive expressions of the static and dynamic self-energy up to the
third order, by taking into account the set of diagrams required when either
the skeleton or nonskeleton expansions of the single-particle propagator are
assumed. A hierarchy of importance among different diagrams is revealed, and a
particular emphasis is given to a third-order diagram (see Fig. 2c) that is
expected to play a significant role among those featuring an
interaction-irreducible three-nucleon force. A consistent formalism to resum at
infinite order correlations induced by three-nucleon forces in the
self-consistent Green's function theory is now available and ready to be
implemented in the many-body solvers.
