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

Abstract

I discuss a renormalisable model based on $\Delta(27)$ family symmetry with an $SO(10)$ grand unified theory (GUT) with spontaneous geometrical $CP$ violation. The symmetries are broken close to the GUT breaking scale, yielding the minimal supersymmetric standard model. Low-scale Yukawa structure is dictated by the coupling of matter to $ \Delta(27) $ antitriplets $ \bar{\phi} $ whose vacuum expectation values are aligned in the CSD3 directions by the superpotential. Light physical Majorana neutrinos masses emerge from the seesaw mechanism within $SO(10)$. The model predicts a normal neutrino mass hierarchy with the best-fit lightest neutrino mass $ m_1 \sim 0.3 $ meV, $CP$-violating oscillation phase $\delta^l\approx 280^{\circ}$ and the remaining neutrino parameters all within $1 \sigma$ of their best-fit experimental values.