A_4 modular flavour model of quark mass hierarchies close to the fixed point τ = ω

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 τ = ω≡exp (i 2π /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, g_u and g_d , can be a source of CP violation in addition to the VEV of the modulus, τ = ω + ϵ , (ϵ )^* ϵ , |ϵ |≪ 1 . We find that in the case of real (CP-conserving) g_u and 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 |ϵ | ≅ 0.03 , all other constants being of the same order in magnitude, and correspond approximately to 1: |ϵ |: |ϵ |^2 . The up-type quark mass hierarchies can be achieved with the same |ϵ | ≅ 0.03 but allowing g_u∼𝒪(10) and correspond to 1: |ϵ |/|g_u|: |ϵ |^2/|g_u|^2 . In this setting the reproduction of the value of the CKM element |V_cb| is problematic. A much more severe problem is the correct description of the CP violation in the quark sector since it arises as a higher order correction in ϵ with |ϵ |≪ 1 . We show that this problem is somewhat alleviated in the case of complex g_u and g_d although the rephasing invariant J_CP is larger by a factor of ∼ 1.8 than the correct value. A correct no-fine-tuned description of the quark mass hierarchies, the quark mixing and CP violation is possible with all constants being of the same order in magnitude and complex g_u and g_d , if one allows different values of ϵ in the down-type and up-type quark sectors, or in a modification of the considered model which involves modular forms of level 3 and weights 8, 6 and 4.