$$A_4$$ A 4 modular flavour model of quark mass hierarchies close to the fixed point

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 $$A_4$$ A 4 symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $$\tau = \omega \equiv \exp ({i\,2\pi /3})$$ τ = ω ≡ exp ( i 2 π / 3 ) (the left cusp of the fundamental domain of the modular group), $$\tau $$ 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, $$g_u$$ g u and $$g_d$$ g d , can be a source of CP violation in addition to the VEV of the modulus, $$\tau = \omega + \epsilon $$ τ = ω + ϵ , $$(\epsilon )^* \ne \epsilon $$ ( ϵ ) ∗ ≠ ϵ , $$|\epsilon |\ll 1$$ | ϵ | ≪ 1 . We find that in the case of real (CP-conserving) $$g_u$$ g u and $$g_d$$ g d and common $$\tau $$ τ ( $$\epsilon $$ ϵ ) in the down-type and up-type quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with $$|\epsilon | \cong 0.03$$ | ϵ | ≅ 0.03 , all other constants being of the same order in magnitude, and correspond approximately to $$1: |\epsilon |: |\epsilon |^2$$ 1 : | ϵ | : | ϵ | 2 . The up-type quark mass hierarchies can be achieved with the same $$|\epsilon | \cong 0.03$$ | ϵ | ≅ 0.03 but allowing $$g_u\sim \mathcal{O}(10)$$ g u ∼ O ( 10 ) and correspond to $$1: |\epsilon |/|g_u|: |\epsilon |^2/|g_u|^2$$ 1 : | ϵ | / | g u | : | ϵ | 2 / | g u | 2 . In this setting the reproduction of the value of the CKM element $$|V_{\textrm{cb}}|$$ | 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 $$\epsilon $$ ϵ with $$|\epsilon |\ll 1$$ | ϵ | ≪ 1 . We show that this problem is somewhat alleviated in the case of complex $$g_u$$ g u and $$g_d$$ g d although the rephasing invariant $$J_{\textrm{CP}}$$ J CP is larger by a factor of $$\sim 1.8$$ ∼ 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 $$g_u$$ g u and $$g_d$$ g d , if one allows different values of $$\epsilon $$ 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.