Constraints on CP violation in the Higgs sector from the $\rho$ parameter

Abstract

We discuss the relation between the CP symmetry and the custodial $SU(2)$ symmetry in the Higgs sector. In particular, we show that CP violation in the Higgs-gauge sector is allowed only if the custodial $SU(2)$ symmetry is broken. We exploit these facts to constrain CP violation using the experimental bounds on $\rho-1$. CP nonconservation in the Higgs-fermion interactions can also be constrained in a similar way although a possible exception is pointed out.