Is there a dynamical group structure behind the bilarge form of neutrino mixing matrix?

Abstract

We observe that the {\it invariance} of neutrino mixing matrix under the simultaneous discrete transformations $ν_1, ν_2, ν_3 \to -ν_1, -ν_2, ν_3 $ and $ν_e, ν_μ, ν_τ\to -ν_e, ν_τ, ν_μ$ (neutrino "horizontal conjugation") {\it characterizes} (as a sufficient condition for it) the familiar bilarge form of neutrino mixing matrix, favored experimentally at present. Thus, the mass neutrinos $ν_1, ν_2, ν_3 $ get a new quantum number, {\it covariant} with respect to their mixings into the flavor neutrinos $ν_e, ν_μ, ν_τ$ (neutrino "horizontal parity" equal to -1, -1,1, respectively). The "horizontal parity" turns out to be embedded in a group structure consisting of some Hermitian and real $3\times 3$ matrices $μ_1, μ_2, μ_3 $ and $ϕ_1, ϕ_2, ϕ_3 $, forming pairs interconnected through neutrino mixings. They generate some discrete transformations of mass and flavor neutrinos, respectively, in such a way that the group relations $μ_1 μ_2 = μ_3 $ (cyclic) and $ϕ_1 ϕ_2 = ϕ_3 $ (cyclic) hold, while $μ_a μ_b = μ_b μ_a $ and $ϕ_a ϕ_b = ϕ_b ϕ_a $. Then, for instance, the $μ_3$ matrix may be chosen equal to the "horizontal parity".