Origin of Symmetric PMNS and CKM Matrices

TL;DR

This paper explores the theoretical origins of the near-symmetry in PMNS and CKM matrices, identifying geometric properties of discrete flavor groups that produce symmetric mixing matrices, and demonstrates a specific symmetric lepton mixing scheme.

Contribution

It identifies geometric properties of discrete flavor symmetry groups that lead to symmetric mixing matrices and applies this to generate a specific symmetric lepton mixing scheme.

Findings

Symmetric mixing matrices are common in groups like A4, S4, Δ(96).

A theorem links geometric properties of groups to symmetric mixing.

A specific symmetric lepton mixing scheme is constructed using Δ(96).

Abstract

The PMNS and CKM matrices are phenomenologically close to symmetric, and a symmetric form could be used as zeroth-order approximation for both matrices. We study the possible theoretical origin of this feature in flavor symmetry models. We identify necessary geometric properties of discrete flavor symmetry groups that can lead to symmetric mixing matrices. Those properties are actually very common in discrete groups such as $A_{4}$, $S_{4}$ or $\Delta(96)$. As an application of our theorem, we generate a symmetric lepton mixing scheme with $\theta_{12}=\theta_{23}=36.21^{\circ};\theta_{13}=12.20^{\circ}$ and $\delta=0$, realized with the group $\Delta(96)$.