Maximal $CP$ violation in the Higgs Sector and its Effect on the $\rho$ Parameter

We study the conditions of maximal $CP$ violation in the neutral Higgs mass matrix of the two Higgs doublet model. We get fixed values of $\tan\b$ and constraints on the Higgs potential parameters. Two neutral Higgs scalars are constrained to be lighter than the charged Higgs scalar and these two Higgs scalars are expected to be almost degenerate due to the smallness of the $h$ parameter, where $h$ is the CP violating coupling constant of the Higgs interaction. The radiative correction of the $\rho$ parameter from the Higgs scalar exchange is rather small and its sign negative for a wide range of Higgs masses. It follows that maximum $CP$ violation in the two Higgs doublet model is safely allowed for the $\rho$ parameter without the custodial symmetry.