  We first show that the many-body Hamiltonian governing the physical
properties of an alkaline-earth Yb-173 Fermi gas across the recently-realized
orbital Feshbach resonance is exactly analogous to that of two-band s-wave
superconductors with contact interactions: i.e., even though the free-particle
bands have a tunable energy offset in between and are coupled by a
Josephson-type attractive inter-band pair scattering, the intra-band
interactions have exactly the same strength. We then introduce two intra-band
order parameters within the BCS mean-field approximation, and investigate the
competition between their in-phase and out-of-phase (i.e., the so-called
\pi-phase) solutions in the entire BCS-BEC evolution at zero temperature.
