Towards a complete \(\Delta(27) \times SO(10)\) SUSY GUT

Abstract

We propose a renormalisable model based on \(\Delta(27)\) family symmetry with an \(SO(10)\) grand unified theory (GUT) leading to a novel form of spontaneous geometrical CP violation. The symmetries, including \(\Delta(27)\) and \(\mathbb{Z}_{9} \times \mathbb{Z}_{12} \times \mathbb{Z}_{4}^{R}\), are broken close to the GUT breaking scale to yield the minimal supersymmetric standard model (MSSM) with the standard R-parity. \(SO(10)\) is broken via \(SU(5)\) with doublet-triplet splitting achieved by a version of the Dimopoulos-Wilczek (missing VEV) mechanism. Low-scale Yukawa structure is dictated by the coupling of matter to \( \Delta(27) \) antitriplets \( \bar{\phi} \) whose VEVs are aligned in the CSD3 directions by the superpotential. Light physical Majorana neutrinos masses emerge from a specific implementation of the seesaw mechanism within \(SO(10)\). The model predicts a normal neutrino mass hierarchy with the best-fit lightest neutrino mass \( m_1 = 0.13 \) meV and the PMNS mixing parameters \(\theta^l_{13} \approx 8.35^{\circ}\), \(\theta^l_{12}\approx 34^{\circ}\), \(\theta^l_{23}\approx 43^{\circ}\) and an oscillation phase \(\delta^l\approx -90^{\circ}\).