$$S^\prime _4$$ 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 $$S^\prime _4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> <mml:mo>′</mml:mo> </mml:msubsup> </mml:math> modular symmetry having minimal number of parameters. The model is considered in the vicinity of the fixed point $$\tau _\text {T}= i\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>T</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>i</mml:mi> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , $$\tau _\text {vev} \sim \tau _\text {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>vev</mml:mtext> </mml:msub> <mml:mo>∼</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>T</mml:mtext> </mml:msub> </mml:mrow> </mml:math> , $$\tau _\text {vev}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>vev</mml:mtext> </mml:msub> </mml:math> being the vacuum expectation value (VEV) of the modulus $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> , 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 $$\tau _\text {vev}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>vev</mml:mtext> </mml:msub> </mml:math> 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 $$|V_{ub}|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>ub</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$|V_{cb}|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>cb</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> elements of the CKM matrix and, in the case of a very high scale of supersymmetry breaking, with the current “average” experimental values of $$|V_{ub}|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>ub</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$|V_{cb}|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>cb</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .