Unitarity boomerangs of quark and lepton mixing matrices

TL;DR

This paper explores unitarity boomerangs as a novel way to visualize quark and lepton mixing matrices, comparing different parametrizations and extending the concept to the lepton sector for a comprehensive understanding.

Contribution

It introduces the unitarity boomerang concept to both quark and lepton mixing matrices and compares its application in KM and CK parametrizations.

Findings

Unitarity boomerangs effectively visualize all four parameters of the KM matrix.

The study compares unitarity boomerangs in KM and CK parametrizations.

Extension of unitarity boomerangs to the lepton sector is feasible and insightful.

Abstract

The most popular way to present mixing matrices of quarks (CKM) and leptons (PMNS) is the parametrization with three mixing angles and one CP-violating phase. There are two major options in this kind of parametrizations, one is the original Kobayashi-Maskawa (KM) matrix, and the other is the Chau-Keung (CK) matrix. In a new proposal by Frampton and He, a unitarity boomerang is introduced to combine two unitarity triangles, and this new presentation displays all four independent parameters of the KM parametrization in the quark sector simultaneously. In this paper, we study the relations between KM and CK parametrizations, and also consider the quark-lepton complementarity (QLC) in the KM parametrization. The unitarity boomerang is discussed in the situation of the CK parametrization for comparison with that in the KM parametrization in the quark sector. Then we extend the idea of unitarity boomerang to the lepton sector, and check the corresponding unitarity boomerangs in the two cases of parametrizations.