Basis invariant measure of CP-violation and renormalization

We analyze, in the context of a simple toy model, for which renormalization schemes the CP-properties of bare Lagrangian and its finite part coincide. We show that this is the case for the minimal subtraction and on-shell schemes. The CP-properties of the theory can then be characterized by CP-odd basis invariants expressed in terms of renormalized masses and couplings. For the minimal subtraction scheme we furthermore show that in CP-conserving theories the CP-odd basis invariants are zero at any scale but are not renormalization group invariant in CP-violating ones.