Origin of Symmetric PMNS and CKM Matrices

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 $\Delta(96)$.