Embedding the neutrino four-group Z2 x Z2 into the alternating group of four objects A4
Wojciech Krolikowski
Abstract
If and only if the small neutrino mixing sine s13 is put zero, the effective neutrino mass matrix M with any mixing sines s12 and s23 is invariant under a cyclic group Z2 of the order two and -- in the limit of m2 - m1 --> 0 -- also under the group Z2 x Z2 often called the four-group. However, the elements of this group do not build up fully the term of M proportional to m2 - m1. When the four-group is embedded into the group of even permutations of four objects A4 (which is of the order twelve), then the term of M proportional to m2 - m1 can be fully constructed from elements of A4. In contrast, when the four-group is embedded into the dihedral group D4 of the order eight, then the term of M proportional to m2 - m1 cannot be fully built up from elements of D4.
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