Neutrino masses and mixings with an S 3 family permutation symmetry

Large neutrino mixing angles suggest that the Yukawa sector is invariant under permutations of the fermion families. This S3 permutation symmetry is broken at a large energy scale but much below the unification scale. Assuming that the lepton mass matrix is approximately diagonal, all neutrino mixing angles naturally come from the breaking of S3 →S2. In the neutrino sector, S2 remains (approximately) unbroken. As a consequence, we have a large atmospheric neutrino angle and Ue3 = 0. The S3 symmetry at the unification scale can also explain the large solar mixing angle. We give an explicit expression of the solar mixing angle in terms of the left-handed neutrino masses. We observe that this family permutation symmetry comes very naturally from a quantized theory of functionals[1], that is an extension of quantum field theory.