Generalized Jarlskog Invariants, Mass Degeneracies and Echelon Crosses

TL;DR

This paper explores conditions under which CKM matrices exhibit CP-violation, emphasizing the role of higher Jarlskog invariants and introducing the concept of echelon crosses for degenerate cases.

Contribution

It introduces the concept of echelon crosses and establishes that higher Jarlskog invariants are necessary for n≥4 to characterize CP-violation.

Findings

Higher Jarlskog invariants are required for n≥4.

Existence of echelon crosses as a sufficient condition.

Degenerate cases demand additional invariants for characterization.

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.