Is $U_{e3}$ really related to the solar neutrino solutions?

Abstract

It has been said that the measurements of $U_{e3}$ in the lepton flavor mixing matrix would help discriminate between the possible solar neutrino solutions under the natural conditions with the neutrino mass hierarchies of $m_1\ll m_2\ll m_3$ and $m_1\sim m_2\gg m_3$, where $m_i$ is the $i$-th generation neutrino absolute mass. However, it is not true, and the relation between ${\sin }^{2}2{\theta }_{12}$ and $U_{e3}$ obtained by Akhmedov, Branco, and Rebelo is trivial in actual. We show in this paper that the value of $U_{e3}$ cannot predict the solar neutrino solutions without one additional nontrivial condition.

Comments: 7pages, no figure