$$S^\prime _4$$ 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

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 |

.