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

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 $\mathrm{\Delta }{m}_{32}^2\sim 3\times {10}^{-3}$ eV${}^2$ we estimate approximately that $\mathrm{\Delta }{m}_{21}^2\sim \mathcal{O}({10}^{-5}$ eV${}^2)$, what is not very different from the recent KamLAND estimation $\mathrm{\Delta }{m}_{21}^2\sim 7\times {10}^{-5}$ 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.