Embedding the neutrino four-group Z_2 x Z_2 into the alternating group of four objects A_4

Abstract

If and only if the small neutrino mixing sine s_13 is put zero, the effective neutrino mass matrix M with any mixing sines s_12 and s_23 is invariant under a cyclic group Z_2 of the order two and -- in the limit of m_2 - m_1 --> 0 -- also under the group Z_2 x Z_2 often called the four-group. However, the elements of this group do not build up fully the term of M proportional to m_2 - m_1. When the four-group is embedded into the group of even permutations of four objects A_4 (which is of the order twelve), then the term of M proportional to m_2 - m_1 can be fully constructed from elements of A_4. In contrast, when the four-group is embedded into the dihedral group D_4 of the order eight, then the term of M proportional to m_2 - m_1 cannot be fully built up from elements of D_4.