Abstract:We propose a renormalisable model based on A 4 family symmetry with an SU(5) grand unified theory (GUT) which leads to the minimal supersymmetric standard model (MSSM) with a two right-handed neutrino seesaw mechanism. Discrete Z 9 × Z 6 symmetry provides the fermion mass hierarchy in both the quark and lepton sectors, while Z R 4 symmetry is broken to Z R 2 , identified as usual R-parity. Proton decay is highly suppressed by these symmetries. The strong CP problem is solved in a similar way to the Nelson-Barr mechanism. We discuss both the A 4 and SU(5) symmetry breaking sectors, including doublet-triplet splitting, Higgs mixing and the origin of the µ term. The model provides an excellent fit (better than one sigma) to all quark and lepton (including neutrino) masses and mixing with spontaneous CP violation. With the A 4 vacuum alignments, (0, 1, 1) and (1, 3, 1), the model predicts the entire PMNS mixing matrix with no free parameters, up to a relative phase, selected to be 2π/3 from a choice of the nine complex roots of unity, which is identified as the leptogenesis phase. The model predicts a normal neutrino mass hierarchy with leptonic angles θ l 13 ≈ 8.7 • , θ l 12 ≈ 34 • , θ l 23 ≈ 46 • and an oscillation phase δ l ≈ −87 • .

We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric SU (5) in 6d, where the two extra dimensions are compactified on a T 2 /Z 2 orbifold. We have shown that, if there is a finite modular symmetry, then it can only be A 4 with fixed modulus τ = ω = e i2π/3 or τ = ω + 1, where we focus on the first possibility. The fields on the branes respect a generalised CP and flavour symmetry A 4 Z 2 which leads to an effective µ − τ reflection symmetry at low energies, implying maximal atmospheric mixing and maximal leptonic CP violation. We construct an explicit model along these lines with two triplet flavons in the bulk, whose vacuum alignments are determined by orbifold boundary conditions, analogous to those used for SU (5) breaking with doublet-triplet splitting. There are two right-handed neutrinos on the branes whose Yukawa couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark Yukawa matrix with high modular weight to provide quark CP violation.

We propose a renormalisable model based on ∆(27) family symmetry with an SO(10) grand unified theory (GUT) leading to a novel form of spontaneous geometrical CP violation. The symmetries, including ∆(27) and Z 9 × Z 12 × Z R 4 , 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 ∆(27) antitriplets φ 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 between 0.32−0.38 meV, CP-violating oscillation phase δ l ≈ (275−280) • and the remaining neutrino parameters all within 1σ of their best-fit experimental values.

We estimate the Baryon Asymmetry of the Universe (BAU) arising from leptogenesis within a class of minimal predictive seesaw models involving two right-handed neutrinos and simple Yukawa structures with one texture zero. The two righthanded neutrinos are dominantly responsible for the "atmospheric" and "solar" neutrino masses with Yukawa couplings to (ν e , ν µ , ν τ ) proportional to (0, 1, 1) and (1, n, n−2), respectively, where n is a positive integer. The neutrino Yukawa matrix is therefore characterised by two proportionality constants with their relative phase providing a leptogenesis-PMNS link, enabling the lightest right-handed neutrino mass to be determined from neutrino data and the observed BAU. We discuss an SU (5) SUSY GUT example, where A 4 vacuum alignment provides the required Yukawa structures with n = 3, while a Z 9 symmetry fixes the relatives phase to be a ninth root of unity.