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An ab-initio calculation scheme for finite nuclei based on self-consistentGreen's functions in the Gorkov formalism is developed. It aims at describingproperties of doubly-magic and semi-magic nuclei employing state-of-the-artmicroscopic nuclear interactions and explicitly treating pairing correlationsthrough the breaking of U(1) symmetry associated with particle numberconservation. The present paper introduces the formalism, necessary toundertake applications at (self-consistent) second-order using two-nucleoninteractions, in a detailed and self-contained fashion. First applications ofsuch a scheme will be reported soon in a forthcoming publication. Future workswill extend the present scheme to include three-nucleon interactions andimplement more advanced truncation schemes.

Ab-initio self-consistent Gorkov-Green's function calculations of semi-magic nuclei - I. Formalism at second order with a two-nucleon interaction