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

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

Comments: 36 pages, no figure. arXiv admin note: text overlap with arXiv:2102.07488