  The two-dimensional Rashba-Hubbard model is investigated in order to clarify
the electron correlation effects in non-centrosymmetric metals. The
renormalization effect on Rashba spin-orbit coupling (RSOC) is calculated on
the basis of second-order and third-order perturbation theories. We show that
RSOC is enhanced by the electron correlation. On the other hand, the
spin-splitting of the Fermi momentum (SFM) is robust against the electron
correlation effect because the enhancement of RSOC is cancelled by the $k$-mass
renormalization. The cancellation is almost perfect in often adopted models in
which the momentum dependence of RSOC is linear in the quasiparticle velocity.
A small correction to the SFM appears otherwise. We show that the same results
are obtained for other antisymmetric spin-orbit couplings in
non-centrosymmetric systems, such as the synthetic spin-orbit coupling in cold
atoms.
