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.
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