$S^\prime_4$ Quark Flavour Model in the Vicinity of the Fixed Point $τ= i\infty$

TL;DR

This work develops a minimal $S^\prime_4$ modular-flavour model for quarks, analyzed near the fixed point $\tau_T=i\infty$, where quark mass hierarchies emerge from small modular-parameter expansions. CP violation is introduced explicitly via a complex down-sector parameter $g_d$, with the modulus $\tau$ playing a suppressed role in CPV, enabling a predictive analytic treatment of the quark mass matrices and CKM matrix. Numerical fits to CKM data across RG scales show the model is viable for inclusive and high-scale SUSY scenarios, while exclusive data are less compatible, highlighting the sensitivity to input data choices. The results support modular symmetry as a promising framework to address the SM flavor problem with a small, well-constrained parameter set and analytic insight into the origin of quark masses and CP violation.

Abstract

We study in the bottom-up framework the possibility to generate the quark mass hierarchies without fine-tuning, the quark mixing and CP-violation (CPV) in a flavour model with $S^\prime_4$ modular symmetry having minimal number of parameters. The model is considered in the vicinity of the fixed point $τ_\text{T}= i\infty$, $τ_\text{vev} \sim τ_\text{T}$, $τ_\text{vev}$ being the vacuum expectation value (VEV) of the modulus $τ$, which allows to explain the hierarchies of the quark masses. The ten quark observables are described 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 $τ_\text{vev}$ 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 $|V_{ub}|$ and $|V_{cb}|$ elements of the CKM matrix and, in the case of a very high scale of supersymmetry breaking, with the current ``average'' experimental values of $|V_{ub}|$ and $|V_{cb}|$.