S^ ₄ quark flavour model in the vicinity of the fixed point τ = i∞

We study in the bottom-up framework the possibility to generate the quark mass hierarchies without severe fine-tuning, the quark mixing and CP-violation (CPV) in a flavour model with S^ ₄ S 4 ′ modular symmetry having minimal number of parameters. The model is considered in the vicinity of the fixed point τ T= i∞ τ T = i ∞ , τ ᵥev ~ τ T τ vev ∼ τ T , τ ᵥev τ vev being the vacuum expectation value (VEV) of the modulus τ τ , which allows to explain the hierarchies of the quark masses. The ten quark observables are described effectively by nine real parameters. The CP-symmetry is broken explicitly since, as is well known, reproducing the observed CPV in the quark sector in the case of spontaneous breaking of CP-symmetry by τ ᵥev τ vev is highly problematic in the class of minimal modular quark flavour models (explaining the quark mass hierarchies without fine-tuning) of the type we consider. We perform a statistical analysis of the model and show that it is phenomenologically viable and consistent, in particular, with the “inclusive” decay data on the |Vub| | V ub | and |Vcb| | V cb | elements of the CKM matrix and, in the case of a very high scale of supersymmetry breaking, with the current “average” experimental values of |Vub| | V ub | and |Vcb| | V cb | .