Neutrino Mass Matrix from ∆(27) Symmetry

The discrete subgroup ∆(27) of SU(3) has some interesting properties which may be useful for understanding charged-lepton and neutrino mass matrices. Assigning leptons to the 3 and 3̄ representations of ∆(27), a simple form of the Majorana neutrino mass matrix is obtained and compared to present data. Since the introduction of the discrete symmetry A4 [1, 2] for understanding the family structure of leptons, much progress has been made [3] in obtaining the so-called tribimaximal mixing pattern of Harrison, Perkins, and Scott [4]. Whereas A4 is the group of even permutation of four objects, it is also the symmetry group of the perfect tetrahedron [5], and identical to the subgroup ∆(12) of SU(3). The next subgroup in the series ∆(3n) is ∆(27), which has recently been applied [6] to quark and lepton mass matrices. In this note, a minimal alternative for leptons is proposed, which results in a Majorana neutrino mass matrix of the form Mν =     fa c b c fb a b a fc     , (1) in the basis where Ml is diagonal. For comparison, two previously proposed models based on A4 have [7] Mν =     a d d d b d d d c     , (2) and [8] Mν =     fa √ ab √ ac √ ab fb √ bc √ ac √ bc fc     , (3) respectively. All three mass matrices have four complex parameters and three predictions, but they are different. In the limit b = c [9], there are five predictions, three of which are common to all three models, i.e. θ23 = π/2, θ13 = 0, and the CP nonconserving Dirac phase is zero (because θ13 = 0). The non-Abelian discrete group ∆(27) has 27 elements divided into 11 equivalence classes. It has 9 one-dimensional irreducible representations 1i(i = 1, ..., 9) and 2 three-dimensional ones 3 and 3̄. Its character table is given below, where n is the number of elements, h is the order of each element, and ω = exp(2πi/3) with 1 + ω + ω = 0. 2 Table 1: Character table of ∆(27). Class n h 11 12 13 14 15 16 17 18 19 3 3̄ C1 1 1 1 1 1 1 1 1 1 1 1 3 3 C2 1 3 1 1 1 1 1 1 1 1 1 3ω 3ω 2 C3 1 3 1 1 1 1 1 1 1 1 1 3ω 2 3ω C4 3 3 1 ω ω 2 1 ω ω 1 ω ω 0 0 C5 3 3 1 ω 2 ω 1 ω ω 1 ω ω 0 0 C6 3 3 1 1 1 ω 2 ω ω ω ω ω 0 0 C7 3 3 1 ω ω 2 ω ω 1 ω ω 1 0 0 C8 3 3 1 ω 2 ω ω 1 ω ω 1 ω 0 0 C9 3 3 1 1 1 ω ω ω ω 2 ω ω 0 0 C10 3 3 1 ω 2 ω ω ω 1 ω ω 1 0 0 C11 3 3 1 ω ω 2 ω 1 ω ω 1 ω 0 0 The group multiplication rules are 3 × 3 = 3̄ + 3̄ + 3̄, and 3 × 3̄ = 9 ∑ i=1 1i, (4) where 11 = 11̄ + 22̄ + 33̄, 12 = 11̄ + ω22̄ + ω 33̄, 13 = 11̄ + ω 22̄ + ω33̄, (5) 14 = 12̄ + 23̄ + 31̄, 15 = 12̄ + ω23̄ + ω 31̄, 16 = 12̄ + ω 23̄ + ω31̄, (6) 17 = 21̄ + 32̄ + 13̄, 18 = 21̄ + ω 32̄ + ω13̄, 19 = 21̄ + ω32̄ + ω 13̄. (7) Let the lepton doublets (νi, li) transform as 3 under ∆(27) and the lepton singlets l c i as 3̄, then with three Higgs doublets transforming as 11, 12, 13, the charged-lepton mass matrix is diagonal and has three independent masses. At the same time, with three Higgs triplets transforming as 3, the form of Eq. (1) is obtained. To see this, consider the product 3× 3 × 3. From Eq. (4), it is clear that it contains three ∆(27) invariants, i.e. 123 + 231 + 312− 213− 321− 132 [which is invariant under SU(3)], 123 + 231 + 312 + 213 + 321 + 132 3 [which is also invariant under A4], and 111 + 222 + 333. Since a Majorana mass matrix has to be symmetric, only the latter two are allowed. Without any loss of generality, the vacuum expectation values of ξ 1,2,3 may then be taken in the proportion a : b : c. Given the form of Eq. (1), the limit θ13 = 0 requires b = c. Under this latter assumption and rotating to the basis [νe, (νμ + ντ )/ √ 2, (−νμ + ντ )/ √ 2], Eq. (1) becomes Mν = 