Bilarge mixing matrix and its invariance under "horizontal conjugation" -- a new discrete transformation for neutrinos

Abstract

In the first part of the note, we consider a neutrino texture, where the Dirac and righthanded Majorana masses are proportional. If the former are approximately proportional also to the charged lepton masses, then taking $\Delta m^2_{32}\sim3\times 10^{-3}$ eV$^2$ we estimate approximately that $\Delta m^2_{21}\sim{\cal O}(10^{-5}$ eV$^2)$, what is not very different from the recent KamLAND estimation $\Delta m^2_{21}\sim7\times 10^{-5}$ eV$^2$, consistent with the LMA solar solution. In the second part, we show generically that the {\it invariance} of neutrino mixing matrix under the simultaneous discrete transformations $\nu_e\to-\nu_e$, $\nu_\mu\to\nu_\tau$, $\nu_\tau\to\nu_\mu$ and $\nu_1\to-\nu_1$, $\nu_2\to-\nu_2$, $\nu_3\to\nu_3$ (neutrino "horizontal conjugation") {\it characterizes} the familiar bilarge form of mixing matrix, favored phenomenologically at present. Then, in the case of this form, the mass neutrinos $\nu_1$, $\nu_2$, $\nu_3$ get a new quantum number, {\it covariant} in their mixings (neutrino "horizontal parity", equal to $-1,-1, 1$, respectively). Conversely, such a covariance may be the {\it origin} of the bilarge mixing matrix. In Section 5, the "horizontal parity" is embedded in a group structure.