Towards a complete ∆(27) × SO(10) SUSY Grand Unified Theory

Towards a complete ∆(27) × SO(10) SUSY Grand Unified Theory I discuss a renormalisable model based on ∆(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 ∆(27) antitriplets φ
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 m1 ∼ 0.3
meV, CP-violating oscillation phase δ
l ≈ 280◦ and the remaining neutrino
parameters all within 1σ of their best-fit experimental values. Introduction
It is well established that the Standard Model (SM) remains incomplete while it fails
to explain why neutrinos have mass. Small Dirac masses may be added by hand, but
this gives no insight into the Yukawa couplings of fermions to Higgs (where a majority
of free parameters in the SM originate), or the extreme hierarchies in the fermion mass
spectrum, ranging from neutrino masses of O(meV) to a top mass of O(100) GeV.
Understanding this, and flavour mixing among quarks and leptons, constitutes the
flavour puzzle. Other open problems unanswered by the SM include the sources of
CP violation (CPV), as well as the origin of three distinct gauge forces, and why
they appear to be equal at very high energy scales.
An approach to solving these puzzles is to combine a Grand Unified Theory (GUT)
with a family symmetry which controls the structure of the Yukawa couplings. In the
highly attractive class of models based on SO(10) [1] , three right-handed neutrinos
are predicted and neutrino mass is therefore inevitable via the seesaw mechanism.
In this paper I summarise a recently proposed model [2], renormalisable at the
GUT scale, capable of addressing all the above problems, based on ∆(27) × SO(10).