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.

Comments: 8 pages, LaTex, This paper updates an earlier version by incorporating KamLAND data