Generalized Cp Invariance, Neutral Flavor Conservation and the Structure of the Mixing Matrix

CP invariance of SU (2) L × U (1) gauge theories is formulated in the most general way. Certain useful standard forms for CP transformations are presented. After a review of horizontal symmetries inducing neutral flavor conservation, we discuss a toy model where CP invariance enforces neutral flavor conservation in a nontrivial way. It is shown that spontaneous CP violation in its general form and neutral flavor conservation lead to a real mixing matrix for three generations. For more than three generations, the mixing matrix may violate CP. A complete analysis of the possible mixing matrices for any number of generations is performed for the usual case of real Yukawa couplings. If the mixing matrix violates CP, some of its elements must be equal in absolute magnitude. For four generations, all possible structures compatible with experiment are specified.