  An ab-initio calculation scheme for finite nuclei based on self-consistent
Green's functions in the Gorkov formalism is developed. It aims at describing
properties of doubly-magic and semi-magic nuclei employing state-of-the-art
microscopic nuclear interactions and explicitly treating pairing correlations
through the breaking of U(1) symmetry associated with particle number
conservation. The present paper introduces the formalism, necessary to
undertake applications at (self-consistent) second-order using two-nucleon
interactions, in a detailed and self-contained fashion. First applications of
such a scheme will be reported soon in a forthcoming publication. Future works
will extend the present scheme to include three-nucleon interactions and
implement more advanced truncation schemes.
