Interactions of ultracold atoms with Rashba spin-orbit coupling, now studied with simulated gauge fields, have non-trivial ultraviolet and infrared behavior. We show that for both physically realizable bosonic and fermionic systems possible ultraviolet logarithmic divergences are absent. We calculate exactly the leading order effective interaction and elucidate the relation between mean-field interactions and free particle scattering lengths, and find infrared logarithmic divergences in the two particle propagator which cause the effective interaction to vanish at zero center-of-mass momentum.