A Geometric Parametrization of the Cabibbo-Kobayashi-Maskawa Matrix and the Jarlskog Invariant

TL;DR

This paper introduces a geometric parametrization of the CKM matrix and Jarlskog invariant using flag manifolds, extending the approach to include a hypothetical fourth quark generation for CP violation analysis.

Contribution

It presents a novel geometric framework for the CKM matrix and Jarlskog invariant, generalized to four quark generations.

Findings

Geometric parametrization using $SU(3)/U(1)^2$ for three generations.

Extension to $SU(4)/U(1)^3$ for four generations.

Provides a new perspective on CP violation analysis.

Abstract

In this paper we give a geometric parametrization to the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix and the Jarlskog invariant, which is based on two flag manifolds $SU(3)/U(1)^{2}$. To treat a fourth generation of quarks on CP violation we generalize the parametrization to one based on two flag manifolds $SU(4)/U(1)^{3}$.