$Z_3$ symmetry of the CKM and PMNS matrices

Abstract

We develop for the CKM and PMNS matrices a new representation with special properties. It is obtained by splitting each of these matrices into two rotations by the angle ${\sim}2\pi/3$ and a universal diagonal matrix with elements, which are cubic roots of~1. Such a representation of the CKM and PMNS matrices may indicate the $Z_{3}$ symmetry to be present in the Yukawa sector of the~SM. Identical mathematical structure of the CKM and PMNS matrices is also an extension of the quark-lepton universality. In this approach the CP violation is a natural consequence of the structure of the Yukawa couplings. The CP violating phase is not a fitted parameter and its value is governed by the parameters of two rotations. The parameters of the diagonalizing matrices of the bi-unitary transformation do not exhibit a hierarchy, which means that the origins of the hierarchy of quark masses and of the CKM matrix elements are not the same.