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

We propose a renormalisable model based on $Δ(27)$ family symmetry with an $SO(10)$ grand unified theory (GUT) leading to a novel form of spontaneous geometrical CP violation. The symmetries, including $Δ(27)$ 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 $Δ(27)$ antitriplets $\barϕ$ 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 $θ^l_{13} \approx 8.35^{\circ}$, $θ^l_{12}\approx 34^{\circ}$, $θ^l_{23}\approx 43^{\circ}$ and an oscillation phase $δ^l\approx -90^{\circ}$.

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