CP violation of quarks in A4 modular invariance

Abstract

We discuss the quark mass matrices in the A4 modular symmetry, where the A4 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 τ. By inputting six quark masses and three CKM mixing angles, we can predict the CP violation phase δ and the Jarlskog invariant JCP. The predicted ranges of δ and JCP are consistent with the observed values. The absolute value of Vub is smaller than 0.0043, while Vcb is larger than 0.0436. In conclusion, our quark mass matrices with the A4 modular symmetry can reproduce the CKM mixing matrix completely with observed quark masses. Keywords: Modular group, A4 non-Abelian discrete symmetry, Quark sector