Constructing the off-diagonal part of active-neutrino mass matrix from annihilation and creation matrices in neutrino-generation space

Abstract

The off-diagonal part of the active-neutrino mass matrix is constructed from two $3\times 3$ matrices playing the role of annihilation and creation matrices acting in the neutrino-generation space of $ν_e, ν_μ, ν_τ$. The construction leads to a new relation, $M_{μτ} = 4\sqrt{3} M_{e μ} $, which predicts in the case of tribimaximal neutrino mixing that $m_3 - m_1 = η(m_2 - m_1)$ with $η= 5.28547$. Then, the maximal possible value of ${Δm^2_{32}}/{Δm^2_{21}}$ is equal to $η^2 -1 = 26.9362$ and gives $m_1 = 0$. With the experimental estimate ${Δm^2_{21}}\sim 8.0\times 10^{-5} {\rm eV}^2$, this maximal value, if realized, predicts $Δm^2_{32} \sim 2.2\times 10^{-3} {\rm eV}^2$, near to the popular experimental estimation $Δm^2_{32} \sim 2.4\times 10^{-3} {\rm eV}^2$.