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Background. The Gorkov approach to self-consistent Green's function theoryhas been formulated in [V. Som\`a, T. Duguet, C. Barbieri, Phys. Rev. C 84,064317 (2011)]. Over the past decade, it has become a method of reference forfirst-principle computations of semi-magic nuclear isotopes. The currentlyavailable implementation is limited to a second-order self-energy and neglectsparticle-number non-conserving terms arising from contracting three-particleforces with anomalous propagators. For nuclear physics applications, this issufficient to address first-order energy differences, ground-state radii andmoments on an accurate enough basis. However, addressing absolute bindingenergies, fine spectroscopic details of $N\pm1$ particle systems or delicatequantities such as second-order energy differences associated to pairing gaps,requires to go to higher truncation orders.  Purpose. The formalism is extended to third order in the algebraicdiagrammatic construction (ADC) expansion with two-body Hamiltonians.  Methods. The expansion of Gorkov propagators in Feynman diagrams is combinedwith the algebraic diagrammatic construction up to the third order as anorganization scheme to generate the Gorkov self-energy.  Results. Algebraic expressions for the static and dynamic contributions tothe self-energy, along with equations for the matrix elements of the Gorkoveigenvalue problem, are derived. It is first done for a general basis beforespecifying the set of equations to the case of spherical systems displayingrotational symmetry. Workable approximations to the full self-consistencyproblem are also elaborated on. The formalism at third order it thus completefor a general two-body Hamiltonian.  Conclusion. Working equations for the full Gorkov-ADC(3) are now availablefor numerical implementation.

Gorkov algebraic diagrammatic construction formalism at third order