Reality-constrained Minimal Yukawa Structure in SO(10) GUT

Abstract

We investigate the minimal Yukawa sector of grand unified theories based on $\mathrm{SO}(10)$ symmetry, consisting of a Higgs structure with representations $\mathbf{10}_{\mathbb{R}}\oplus \mathbf{120}_{\mathbb{R}}\oplus\mathbf{126}$. In this framework, where $\mathbf{10}_\mathbb{R}$ and $\mathbf{120}_{\mathbb{R}}$ are real scalars, we derive the associated $\mathrm{SO}(10)$ reality conditions for their weak-doublet constituents -- both by explicit computation and an analytic reframing into a Pati-Salam-like description -- to revisit previously reported fermion mass relations. Our analysis revises these earlier results, in particular by introducing a relative sign difference between the reality constraints on the two weak doublets in $\mathbf{120}_{\mathbb{R}}$, yielding a new parameter (a magnitude) in the fermion mass relations. Our formalism is fully general and provides a systematic framework for deriving Clebsch-Gordan coefficients and implementing reality constraints for arbitrary parent-daughter representation pairs of $\mathrm{SO}(10)$ and its Pati-Salam subgroup. Incorporating these corrections, we perform an extensive numerical scan of the parameter space and find that the model successfully reproduces SM fermion masses and mixings, including recent precision measurements of solar oscillation parameters by JUNO. It accommodates both octants of $\theta_{23}$ while mildly disfavoring $\delta_\mathrm{PMNS}\sim (140^\circ,220^\circ)$. The model predicts a strongly hierarchical right-handed neutrino spectrum $(10^{5},10^{12},10^{15})$ GeV and a neutrinoless double beta decay parameter $m_{\beta\beta}\sim 3$-$4$ meV, just below future experimental sensitivity. Proton decay is dominated by $p\to\pi^+\overline{\nu}$ and $p\to\pi^0 e^+$, making these channels testable in upcoming experiments.