Neutrino Tribimaximal Mixing from A4 Alone

Neutrino tribimaximal mixing is obtained from the breaking of A4 to Z3 in the charged-lepton sector and to Z2 in the neutrino sector. To enforce this conflicting pattern, extra particles and symmetries are usually invoked, often accompanied by nonrenormalizable interactions and even extra dimensions. It is shown here in a specific renormalizable model how A4 alone will accomplish this, with only the help of lepton number. Introduction: The observed neutrino mixing matrix is very close to the tribimaximal form [1] and is best understood in terms of the tetrahedral symmetry A4 [2, 3, 4, 5]. The key to its success is the pattern of symmetry breaking with preserved subgroups [6, 7] such that A4 → Z3 and A4 → Z2 in two different sectors. This misalignment is technically challenging to achieve and many auxiliary symmetries and particles have been invoked to make it happen [5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], often with nonrenormalizable operators and even in the context of extra dimensions, thereby complicating the simple original idea of using just A4. On closer examination, it turns out that the breaking of Z3 by Z2 through a judicious choice of soft terms in the Higgs potential would work because there is a hidden symmetry which prevents the appearance of the other unwanted terms. This is accomplished in a renormalizable model using Higgs doublets and triplets, both transforming as 1 and 3 of A4 alone. Lepton and Higgs Assignments: The three families of leptons are considered [18] as triplets under A4: Li = (νi, li) ∼ 3, l i ∼ 3. (1) To obtain arbitrary lepton masses me,μ,τ , four Higgs doublets are used: Φ0 = (φ 0 0, φ − 0 ) ∼ 1, Φi = (φ0i , φ−i ) ∼ 3. (2) The Yukawa interactions are then given by LY = h0(ν1φ−0 − l1φ00)lc 1 + h1(ν1φ−3 − l1φ03)lc 2 + h2(ν2φ−3 − l2φ03)lc 1 + (1 → 2, 2 → 3, 3 → 1) + (1 → 3, 3 → 2, 2 → 1) +H.c., (3) resulting in Ml = 