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 $\bar{\varphi }$ 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 ${\theta }_{13}^l\approx 8.35^{\circ }$, ${\theta }_{12}^l\approx 34^{\circ }$, ${\theta }_{23}^l\approx 43^{\circ }$ and an oscillation phase ${\delta }^l\approx -90^{\circ }$.

Comments: 32 pages, 22 figures. v3: revised alignment and fit discussion, typos corrected