| Abstract: |
We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without fine-tuning in a quark flavor model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $\tau=\omega. The model involves modular forms of level 3 and weights 6, 4 and 2, and contains eight constants, only two of which, $g_u$ and $g_d$, can be a source of CP violation in addition to the VEV of the modulus, $\tau = \omega + \epsilon$, $|\epsilon|\ll 1$. We find that the down-type quark mass hierarchies can be reproduced without fine tuning with $|\epsilon| \cong 0.03$, all other constants being same order, and correspond approximately to $1: |\epsilon|: |\epsilon|^2$ by the common $\tau$ in both down-quark and up-quark sectors. The up-type quark mass hierarchies can be achieved with the same $|\epsilon| \cong 0.03$ but allowing one parameter $g_u\si{\cal O}(10)$ and correspond to $1: |\epsilon|/|g_u|: |\epsilon|^2/|g_u|^2$. In this setting the reproduction of the value of the CKM element including CP violation is investigated. Some modifications of the model are also discussed. |
