$A_4$ Modular Flavour Model of Quark Mass Hierarchies close to the Fixed Point $\tau =\omega$

We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $\tau =\omega \equiv \exp (i2\pi \mathrm{/}3)$ (the left cusp of the fundamental domain of the modular group), $\tau$ being the VEV of the modulus. 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, $\tau =\omega +\epsilon$, $(\epsilon {)}^{\ast }\mathrm{\ne }\epsilon$, $\mathrm{\mid }\epsilon \mathrm{\mid }\ll 1$. We find that in the case of real (CP-conserving) $g_u$ and $g_d$ and common $\tau$ ($\epsilon$) in the down-quark and up-quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with $\mathrm{\mid }\epsilon \mathrm{\mid }\cong 0.03$, all other constants being of $\mathcal{O}(1)$, and correspond approximately to $1:\mathrm{\mid }\epsilon \mathrm{\mid }:\mathrm{\mid }\epsilon {\mathrm{\mid }}^2$. The up-type quark mass hierarchies can be achieved with the same $\mathrm{\mid }\epsilon \mathrm{\mid }\cong 0.03$ but allowing $g_u\sim \mathcal{O}(10)$ and correspond to $1:\mathrm{\mid }\epsilon \mathrm{\mid }\mathrm{/}\mathrm{\mid }{g}_u\mathrm{\mid }:\mathrm{\mid }\epsilon {\mathrm{\mid }}^2\mathrm{/}\mathrm{\mid }{g}_u{\mathrm{\mid }}^2$. In this setting, we discuss the CKM quark mixing and CP violation.