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We propose models in which the hierarchical structures of the masses andmixing in both quark and lepton sectors are explained by the $S_4^\prime$modular flavor symmetry near the fixed point $\tau \sim i\infty$. The modelprovides the first explicit example which explains hierarchies of both quarksand leptons. The hierarchies are realized by powers of $\epsilon = e^{2\pi i\tau/4} = \mathcal{O}(0.01)$ and $2\,\mathrm{Im}\,\tau \sim 5$, where $\tau$being the modulus. The small parameter $\epsilon$ plays a role of flavon in theFroggatt-Nielsen mechanism under the residual $Z_4^T$ symmetry, and powers of$2\,\mathrm{Im}\,\tau$ in the Yukawa couplings are controlled by modularweights via the canonical normalization. The doublet quarks are identified to a$S_4^\prime$ triplet to explain the hierarchical structure of the quark mixingangles, while the doublet leptons are composed of three singlets for the largemixing angles in the lepton sector. We show that the $S_4^\prime$ modularsymmetry alone can explain the hierarchies in both quark and lepton sectors by$\mathcal{O}(1)$ coefficients.

Quark and lepton hierarchies from $S_4^\prime$ modular flavor symmetry