Unitarity constraints onCPnonconservation in Higgs-boson exchange

Bounds are derived for the dimensionless factors that enter in the violation of $\mathrm{CP}$ invariance in Higgs-boson exchange. In particular, for the class of theories with two scalar SU(2)×U(1) doublets [plus any other scalars whose expectation values do not break SU(2)×U(1)], the magnitude of the parameter Im$Z_2$, which appears in the dominant neutral-Higgs-boson-exchange contribution to the neutron electric dipole moment, is bounded by $\frac{1}{2}{(r^2+r^4)}^{\frac{1}{2}}$, where $r$ is a ratio of the magnitudes of scalar vacuum expectation values. It is shown that this bound can actually be reached in realistic models.