# CP violation of quarks in $A_4$ modular invariance

## TL;DR

This paper explores how $A_4$ modular symmetry can model quark mass matrices, successfully reproducing observed CKM mixing and CP violation parameters with a minimal set of parameters.

## Contribution

It introduces a novel $A_4$ modular invariant framework for quark masses that predicts CP violation phases consistent with experimental data.

## Key findings

- Predicted CP phase $oldsymbol{ heta}$ and Jarlskog invariant $oldsymbol{J_{CP}}$ match observed ranges.
- Model constrains CKM matrix elements, notably $V_{ub}$ and $V_{cb}$.
- Reproduces quark masses and mixing angles with six real and two complex parameters.

## Abstract

We discuss the quark mass matrices in the $A_4$ modular symmetry, where the $A_4$ triplet of Higgs is introduced for each up-quark and down-quark sectors, respectively. The model has six real parameters and two complex parameters in addition to the modulus $\tau$. By inputting six quark masses and three CKM mixing angles, we can predict the CP violation phase $\delta$ and the Jarlskog invariant $J_{CP}$. The predicted ranges of $\delta$ and $J_{CP}$ are consistent with the observed values. The absolute value of $V_{ub}$ is smaller than $0.0043$, while $V_{cb}$ is lager than $0.0436$. In conclusion, our quark mass matrices with the $A_4$ modular symmetry can reproduce the CKM mixing matrix completely with observed quark masses.