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 then been included in ab initio calculations on nuclear matter and isotopic chains of finite nuclei.

ractical applications across post Hartree-Fock methods have mostly considered the contribution of three-nucleon interactions in an effective way, 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, including interactionirreducible (i.e. not averaged) three-nucleon diagrams.

We employ the algebraic diagrammatic construction up to third order as the 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 third order, by taking into account also the set of diagrams required when the skeleton expansion of the single-particle propagator is not assumed. A hierarchy of importance among different diagrams is revealed, and a particular emphasis is given to a third-order diagram (see Fig. 2c) which 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. Work is in progress to include the first interaction-irreducible three-nucleon diagram in calculations of closed-shell medium-mass nuclei.