Fermion Masses and Mixing in Pati–Salam Unification with <i>S</i>3 Modular Symmetry

Abstract Modular invariance has recently paved the way for promising new directions in flavor model building. Motivated by this development, we present in this work the first implementation of the $S_3$ modular symmetry within the Pati–Salam unification framework, addressing the flavor structure of quarks and leptons. Assigning left- and right-handed matter fields as $S_3$ doublets or singlets, we propose three benchmark models that achieve compelling fits to 16 observables including charged fermion mass ratios and flavor mixing parameters. Light neutrino masses arise via the type-I seesaw mechanism, and neutrino oscillation parameters are explored in light of the latest NuFIT v6.0 results. All models favor a normal neutrino mass ordering, with the atmospheric mixing angle lying in the lower octant. For models I and III, the effective Majorana mass $m_{\beta \beta }$ is within the reach of upcoming neutrinoless double-beta decay experiments, while it is too small to be detected in model II. Predicted leptonic CP-violating phases and the sum of the active neutrino masses span wide but distinctive ranges, enabling future experiments to test and differentiate the proposed models.