What is the Natural Size of Supersymmetric CP Violation ?

It is well known that if phases and masses in the Minimal Supersymmetric Standard Model (MSSM) are allowed to have general values, the resulting neutron EDM (dn) exceeds the experimental upper limit by about 10 . We assume that the needed suppression is not due to a fine-tuning of phases or masses, and ask what natural size of CP violation (CPV) results. We show that (1) the phase of one of the superpotential parameters, μ, does not contribute to any CPV in the MSSM and so is not constrained by dn; (2) the MSSM contribution to dn is tiny, just coming from the CKM phase; (3) the phases in the MSSM cannot be used to generate a baryon asymmetry at the weak scale, given our assumptions; and (4) in non-minimal SUSY models, an effective phase can enter at one loop giving dn ∼ 10e cm, de ∼ 10e cm, and allowing a baryon asymmetry to be generated at the weak scale, without fine-tunings. Our results could be evaded by a SUSY breaking mechanism which produced phases for the SUSY breaking parameters that somehow were naturally of order 10.