A natural S 4 × SO(10) model of flavour

Abstract We propose a natural S 4 × SO(10) supersymmetric grand unified theory of flavour with an auxiliary ℤ 4 2 × ℤ 4 R $$ {\mathbb{Z}}_4^2\times {\mathbb{Z}}_4^R $$ symmetry, based on small Higgs representations (nothing larger than an adjoint) and hence a type-I seesaw mechanism. The Yukawa structure of all fermions is determined by the hierarchical vacuum expectation values of three S 4 triplet flavons, with CSD3 vacuum alignments, where up-type quarks and neutrinos couple to one Higgs 10, and the down-type quarks and charged leptons couple to a second Higgs 10. The Yukawa matrices are obtained from sums of low-rank matrices, where each matrix in the sum naturally accounts for the mass of a particular family, as in sequential dominance in the neutrino sector, which predicts a normal neutrino mass hierarchy. The model accurately fits all available quark and lepton data, with predictions for the leptonic CP phase in 95% credible intervals given by 281° < δ ℓ < 308° and 225° < δ ℓ < 253°. The model reduces to the MSSM, with the two Higgs doublets emerging from the two Higgs 10s without mixing, and we demonstrate how a μ term of O T e V $$ \mathcal{O}\left(\mathrm{T}\mathrm{e}\mathrm{V}\right) $$ can be realised, as well as doublet-triplet splitting, with Planck scale operators controlled by symmetry, leading to acceptable proton decay