$$S^\prime _4$$ quark flavour model in the vicinity of the fixed point $$\tau = i\infty $$

Abstract 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 $$S^\prime _4$$ S 4 ′ modular symmetry having minimal number of parameters. The model is considered in the vicinity of the fixed point $$\tau _\text {T}= i\infty $$ τ T = i ∞ , $$\tau _\text {vev} \sim \tau _\text {T}$$ τ vev ∼ τ T , $$\tau _\text {vev}$$ τ vev being the vacuum expectation value (VEV) of the modulus $$\tau $$ 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 $$\tau _\text {vev}$$ 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 $$|V_{ub}|$$ | V ub | and $$|V_{cb}|$$ | 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 $$|V_{ub}|$$ | V ub | and $$|V_{cb}|$$ | V cb | .