SpontaneousCPviolation and supersymmetry

One way to solve the strong $\mathrm{CP}$ problem without axions involves $\mathrm{CP}$ being a spontaneously broken symmetry. We discuss certain recent arguments that lend credibility to this idea in the context of superstrings and supersymmetry. We discuss the supersymmetric versions of the so-called Nelson models and show that in them $\overline{\theta }$ arises at the two-loop level. This implies that the neutron electric dipole moment (EDM) should be near the present limit, whereas the electron electric dipole moment should be unobservably small. On the other hand if $\mathrm{CP}$ is not a spontaneously broken symmetry, but explicitly broken, one expects both the neutron and electron EDM's to be not far from the experimental bounds.