Self-consistent Greens 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 Greens function theory is now available and ready to be implemented in the many-body solvers.
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