$$A_4$$ A 4 modular flavour model of quark mass hierarchies close to the fixed point $$\tau = \omega $$ τ = ω

Abstract

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 $$A_4$$ 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.