Residual Z2×Z2Z2×Z2 symmetries and lepton mixing

We consider two novel scenarios of residual symmetries of the lepton mass matrices. Firstly we assume a Z2×Z2Z2×Z2 symmetry GℓGℓ for the charged-lepton mass matrix and a Z2Z2 symmetry GνGν for the light neutrino mass matrix. With this setting, the moduli of the elements of one column of the lepton mixing matrix are fixed up to a reordering. One may interchange the roles of GℓGℓ and GνGν in this scenario, thereby constraining a row, instead of a column, of the mixing matrix. Secondly we assume a residual symmetry group Gℓ≅ZmGℓ≅Zm (m>2m>2) which is generated by a matrix with a doubly-degenerate eigenvalue. Then, with Gν≅Z2×Z2Gν≅Z2×Z2 the moduli of the elements of a row of the lepton mixing matrix get fixed. Using the library of small groups we have performed a search for groups which may embed GℓGℓ and GνGν in each of these two scenarios. We have found only two phenomenologically viable possibilities, one of them constraining a column and the other one a row of the mixing matrix.