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 $\nu_1, \nu_2, \nu_3 \to -\nu_1, -\nu_2, \nu_3 $ and $\nu_e, \nu_\mu, \nu_\tau \to -\nu_e, \nu_\tau, \nu_\mu $ (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 $\nu_1, \nu_2, \nu_3 $ get a new quantum number, {\it covariant} with respect to their mixings into the flavor neutrinos $\nu_e, \nu_\mu, \nu_\tau $ (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 $\mu_1, \mu_2, \mu_3 $ and $\phi_1, \phi_2, \phi_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 $\mu_1 \mu_2 = \mu_3 $ (cyclic) and $\phi_1 \phi_2 = \phi_3 $ (cyclic) hold, while $\mu_a \mu_b = \mu_b \mu_a $ and $\phi_a \phi_b = \phi_b \phi_a $. Then, for instance, the $\mu_3$ matrix may be chosen equal to the "horizontal parity".