  The study of multi-band superconductivity is relevant for a variety of
systems, from ultra cold atoms with population imbalance to particle physics,
and condensed matter. As a consequence, this problem has been widely
investigated bringing to light many new and interesting phenomena. In this work
we point out and explore a correspondence between a two-band metal with a
$k$-dependent hybridization and a uniformly polarized fermionic system in the
presence of spin-orbit coupling (SOC). We study the ground state phase diagram
of the metal in the presence of an attractive interaction. We find remarkable
superconducting properties whenever hybridization mixes orbitals of different
parities in neighboring sites. We show that this mechanism enhances
superconductivity and drives the crossover from weak to strong coupling in
analogy with SOC in cold atoms. We obtain the quantum phase transitions between
the normal and superfluid states, as the intensity of different parameters
characterizing the metal are varied, including Lifshitz transitions, with no
symmetry breaking, associated with the appearance of soft modes in the Fermi
surface.
