Generalized Jarlskog Invariants, Mass Degeneracies and Echelon Crosses

Abstract

It is known that the Cabibbo-Kobayashi-Maskawa (CKM) $n\times n$ matrix can be represented by a real matrix iff there is no CP-violation, and then the Jarlskog invariants vanish. We investigate sufficient conditions for the opposite statement to hold, paying particular attention to degenerate cases. We find that higher Jarlskog invariants are needed for $n\geq 4$. One generic sufficient condition is provided by the existence of a so-called echelon cross.