Residual Z 2 symmetries and leptonic mixing patterns from finite discrete subgroups of U(3)

Abstract

We study embedding of non-commuting Z 2 and Z m , m ≥ 3 symmetries in discrete subgroups (DSG) of U(3) and analytically work out the mixing patterns implied by the assumption that Z 2 and Z m describe the residual symmetries of the neutrino and the charged lepton mass matrices respectively. Both Z 2 and Z m are assumed to be subgroups of a larger discrete symmetry group G f possessing three dimensional faithful irreducible representation. The residual symmetries predict the magnitude of a column of the leptonic mixing matrix U PMNS which are studied here assuming G f as the DSG of SU(3) designated as type C and D and large number of DSG of U(3) which are not in SU(3). These include the known group series Σ(3n 3), T n (m), Δ(3n 2, m), Δ(6n 2, m) and Δ′(6n 2, j, k). It is shown that the predictions for a column of |U PMNS| in these group series and the C and D types of groups are all contained in the predictions of the Δ(6N 2) groups for some integer N. The Δ(6N 2) groups therefore represent a sufficient set of G f to obtain predictions of the residual symmetries Z 2 and Z m .