Littlest Seesaw model from S 4 × U(1)

Abstract

We show how a minimal (littlest) seesaw model involving two right-handed neutrinos and a very constrained Dirac mass matrix, with one texture zero and two independent Dirac masses, may arise from S 4 ×U(1) symmetry in a semi-direct supersymmetric model. The resulting CSD3 form of neutrino mass matrix only depends on two real mass parameters plus one undetermined phase. We show how the phase may be fixed to be one of the cube roots of unity by extending the S 4 × U(1) symmetry to include a product of Z 3 factors together with a CP symmetry, which is spontaneously broken leaving a single residual Z 3 in the charged lepton sector and a residual Z 2 in the neutrino sector, with suppressed higher order corrections. With the phase chosen from the cube roots of unity to be −2π/3, the model predicts a normal neutrino mass hierarchy with m 1 = 0, reactor angle θ 13 = 8.7 • , solar angle θ 12 = 34 • , atmospheric angle θ 23 = 44 • , and CP violating oscillation phase δ CP = −93 • , depending on the fit of the model to the neutrino masses.

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

“…In the limit µ 3 = m 1 = 0, this effectively reduces to a two RH neutrino model, with only two real parameters µ 1 and µ 2 . The resulting model is known as Littlest Seesaw (LS) [29][30][31]. 1 It is remarkable that, with µ 1,2 fitted to the neutrino masses, the entire PMNS matrix is then uniquely determined with no free parameters, giving predictions for mixing angles and the CP phase in agreement with current data.…”

“…It is complete in the sense that we address the µ-problem, Higgs mixing and doublet-triplet splitting, and provide an ultraviolet renormalisable model, with Planck-suppressed operators controlled by symmetry. However, we do not discuss the origin of the hierarchy of flavon VEVs, nor do we repeat the discussion of flavon vacuum alignment, which can be found in [30]. We believe this model represents a signficant step forward in the quest for a complete and correct description of fermions within SUSY GUTs.…”

“…A particularly successful scheme is known as CSD3 [24,[28][29][30][31] where the neutrino Yukawa matrix is controlled by particular vacuum expectation values (VEVs) of three triplet flavon fields, φ i , as discussed later. The flavon vacuum alignments are fixed by a superpotential which we do not specify here, but may be enforced by an S 4 symmetry, as discussed in [30]. After implementing the seesaw mechanism, the above flavons yield a light effective left-handed (LH) Majorana neutrino mass matrix,…”

“…Another important difference is that we have used the CSD3 vacuum alignments in [30], whereas the vacuum aligments used in most previous works were geared towards TB mixing, and do not naturally provide a large reactor angle. For instance, in [38] they predict U e3 ∼ 0.05, which is now excluded.…”

“…The flavon vacuum alignments, obtained from [30], which preserve the generator product SU , are (as in eq. (2.2))…”