Neutrino Mass Matrix and Hierarchy

Abstract

We build a model to describe neutrinos based on strict hierarchy, incorporating as much as possible, the latest known data, for $\Delta_{sol}$ and $\Delta_{atm}$, and for the mixing angles determined from neutrino oscillation experiments, including that from KamLAND. Since the hierarchy assumption is a statement about mass ratios, it lets us obtain all three neutrino masses. We obtain a mass matrix, $M_\nu$ and a mixing matrix, $U$, where both $M_\nu$ and $U$ are given in terms of powers of $\Lambda$, the analog of the Cabibbo angle $\lambda$ in the Wolfenstein representation, and two parameters, $\rho$ and $\kappa$, each of order one. The expansion parameter, $\Lambda$, is defined by $\Lambda^2 = m_2/m_3 = \surd (\Delta_{sol}/\Delta_{atm}) \approx$ 0.16, and $\rho$ expresses our ignorance of the lightest neutrino mass $m_1, (m_1 = \rho \Lambda^4 m_3$), while $\kappa$ scales $s_{13}$ to the experimental upper limit, $s_{13} = \kappa \Lambda^2 \approx 0.16 \kappa$. These matrices are similar in structure to those for the quark and lepton families, but with $\Lambda$ about 1.6 times larger than the $\lambda$ for the quarks and charged leptons. The upper limit for the effective neutrino mass in double $\beta$-decay experiments is $4 \times 10^{-3} eV$ if $s_{13} = 0$ and $6 \times 10^{-3} eV$ if $s_{13}$ is maximal. The model, which is fairly unique, given the hierarchy assumption and the data, is compared to supersymmetric extension and texture zero models of mass generation.