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 (dnβ) exceeds the experimental upper limit by about 103. 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, ΞΌ, does not contribute to
any CPV in the MSSM and so is not constrained by \dn; (2) the MSSM contribution
to dnβ 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 dnββΌ10β26\ecm, deββΌ10β27\ecm, 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 10β3.Comment: 13pp (no figs), REVTEX (LATEX), TRI-PP-93-