Lepton Family Symmetry and Neutrino Mass Matrix

The standard model of leptons is extended to accommodate a discrete Z3 × Z2 family symmetry. After rotating the charged-lepton mass matrix to its diagonal form, the neutrino mass matrix reveals itself as very suitable for explaining atmospheric and solar neutrino oscillation data. A generic requirement of this approach is the appearance of three Higgs doublets at the electroweak scale, with observable flavor violating decays. In the standard model of particle interactions, the fermion mass matrices are a priori completely arbitrary, and yet data seem to indicate specific patterns which are not yet understood theoretically. Recent experimental advances in measuring the neutrino-oscillation parameters in atmospheric [1] and solar [2] data have now fixed the 3 × 3 lepton mixing matrix to a large extent. This information is not sufficient to fix all the elements of the neutrino mass matrix Mν (assumed here to be Majorana at the outset), but is enough to fix its approximate form in terms of a small number of parameters [3]. However, this works only in the basis (νe, νμ, ντ ), i.e. where the charged-lepton mass matrix Ml has been assumed diagonal. If a symmetry is behind the observed pattern of Mν , then it must also apply to Ml. The realization of this in a complete theory of leptons has only been done in a few cases. Some recent examples are those based on the symmetries of geometric objects, i.e. S3 (equilateral triangle) [4], D4 (square) [5], and A4 (tetrahedron) [6, 7]. In this paper, a much simpler and more flexible model is proposed, based on the discrete symmetry Z3 × Z2. [Because of the choice of representations, this turns out to be equivalent to a specific realization of S3.] The key idea which allows Mν to be more restricted than Ml is that they come from different mechanisms. Whereas the former comes from the naturally small vacuum expectation value (vev) of a single heavy Higgs triplet ξ = (ξ, ξ, ξ) [8], thus dispensing with the usual three heavy singlet neutrinos, the latter comes from the vev’s of three Higgs doublets [9, 10]. As shown below, the use of Z3 × Z2 as a lepton family symmetry results in 2 parameters for Mν and 5 parameters for MlM†l . After rotating MlM†l to its diagonal form, i.e. (m2e, m 2 μ, m 2 τ ), then under a condition to be derived below, the neutrino mass matrix will become [10] Mν = 