Horizontal gauge symmetry as a natural CP-violation source at the two-generation level

We show that the classification of the WS-GIM generations by horizontal gauge symmetries can give rise to CP violation already at the two-generation level. The smallness of the CP-violation phase associated with the K⁰-K⁰ system is correlated with the smallness of the physical Cabibbo-like angle which emerges in the up-quark sector. We argue that strict lepton number conservation severely restricts the horizontal symmetries and requires a non-trivial vertical structure. The effective lowest-order four-fermion CP-violating operator is calculated in a recently proposed (SU(2) × U(1)) × U′ (1) model.