How does a synthetic non-Abelian gauge field influence the bound states of two spin-$1/2$ fermions?

We study the bound states of two spin-$1/2$ fermions interacting via a contact attraction (characterized by the scattering length) in the singlet channel in $3D$ space in presence of a uniform non-Abelian gauge field. The configuration of the gauge field that generates a Rashba type spin-orbit interaction is described by three coupling parameters $(\lambda_x, \lambda_y, \lambda_z)$. For a generic gauge field configuration, the critical scattering length required for the formation of a bound state is {\em negative}, i.e., shifts to the ``BCS side'' of the resonance. Interestingly, we find that there are special high-symmetry configurations (e.g., $\lambda_x = \lambda_y = \lambda_z$) for which there is a two body bound state for {\em any} scattering length however small and negative. Our results show that the BCS-BEC crossover is drastically affected by the presence of a non-Abelian gauge field. We discuss possible experimental signatures of our findings both at high and low temperatures.