Quark masses and mixing from Modular $S_4^{\mathrm{\prime }}$ with Canonical Kähler Effects

We propose a quark flavor model based on modular $S_4^{\mathrm{\prime }}$ with a general CP symmetry. CP violation in the quark sector is entirely realized by the modulus $⟨\tau ⟩$. We show that the canonical normalization induced by the Kähler metric plays a crucial role in reproducing the observed hierarchies, while maintaining coupling constants of order $\mathcal{O}(1)$. The minimal model achieves a great fit to the quark sector data, which we take as the PDG 2024 data extrapolated to the GUT scale.