A₄ modular flavour model of quark mass hierarchies close to the fixed point =

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 A₄ A 4 symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point = (i\, 2 /3) τ = ω ≡ exp (i 2 π / 3) (the left cusp of the fundamental domain of the modular group), τ 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, gᵤ g u and gd g d, can be a source of CP violation in addition to the VEV of the modulus, = + τ = ω + ϵ, () ^* (ϵ) ∗ ≠ ϵ, | | 1 | ϵ | ≪ 1. We find that in the case of real (CP-conserving) gᵤ g u and gd g d and common τ (ϵ) in the down-type and up-type quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with | | 0. 03 | ϵ | ≅ 0. 03, all other constants being of the same order in magnitude, and correspond approximately to 1: | |: | |² 1: | ϵ |: | ϵ | 2. The up-type quark mass hierarchies can be achieved with the same | | 0. 03 | ϵ | ≅ 0. 03 but allowing gᵤ O (10) g u ∼ O (10) and correspond to 1: | |/|gᵤ|: | |²/|gᵤ|² 1: | ϵ | / | g u |: | ϵ | 2 / | g u | 2. In this setting