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 $Δ_{sol}$ and $Δ_{atm}$, 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, $M_ν$ and a mixing matrix, $U$, where both $M_ν$ and $U$ are given in terms of powers of $Λ$, the analog of the Cabibbo angle $λ$ in the Wolfenstein representation, and two parameters, $ρ$ and $κ$, each of order one. The expansion parameter, $Λ$, is defined by $Λ^2 = m_2/m_3 = \surd (Δ_{sol}/Δ_{atm}) \approx$ 0.16, and $ρ$ expresses our ignorance of the lightest neutrino mass $m_1, (m_1 = ρΛ^4 m_3$), while $κ$ scales $s_{13}$ to the experimental upper limit, $s_{13} = κΛ^2 \approx 0.16 κ$. These matrices are similar in structure to those for the quark and lepton families, but with $Λ$ about 1.6 times larger than the $λ$ for the quarks and charged leptons. The upper limit for the effective neutrino mass in double $β$-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.