# Toward a unified interpretation of quark and lepton mixing from flavor and CP symmetries

## TL;DR

This paper explores how a combined flavor and CP symmetry framework can predict lepton and quark mixing patterns, providing a unified approach that fits experimental data with minimal free parameters.

## Contribution

It introduces a model where a discrete flavor group and CP symmetry breaking lead to predictive lepton and quark mixing matrices, unifying their description with a minimal group size.

## Key findings

- Predicts lepton mixing angles and CP phases from two free parameters.
- Identifies viable mixing patterns from the group n^2 and CP symmetry.
- Achieves a unified description of quark and lepton mixing with (6n^2) for n=7.

## Abstract

We discussed the scenario that a discrete flavor group combined with CP symmetry is broken to $Z_2\times CP$ in both neutrino and charged lepton sectors. All lepton mixing angles and CP violation phases are predicted to depend on two free parameters $\theta_{l}$ and $\theta_{\nu}$ varying in the range of $[0, \pi)$. As an example, we comprehensively study the lepton mixing patterns which can be derived from the flavor group $\Delta(6n^2)$ and CP symmetry. Three kinds of phenomenologically viable lepton mixing matrices are obtained up to row and column permutations. We further extend this approach to the quark sector. The precisely measured quark mixing angles and CP invariant can be accommodated for certain values of the free parameters $\theta_{u}$ and $\theta_{d}$. A simultaneous description of quark and lepton flavor mixing structures can be achieved from a common flavor group $\Delta(6n^2)$ and CP, and accordingly the smallest value of the group index $n$ is $n=7$.