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Neutrino Masses with "Zero Sum" Condition: $m_{ν_1} + m_{ν_2} + m_{ν_3} = 0$

It is well known that the neutrino mass matrix contains more parameters than experimentalists can hope to measure in the foreseeable future even if we impose CP invariance. Thus, various authors have proposed ansatzes to restrict the form of the neutrino mass matrix further. Here we propose that $m_{ν_1} + m_{ν_2} + m_{ν_3} = 0$; this ``zero sum'' condition can occur in certain class of models, such as models whose neutrino mass matrix can be expressed as commutator of two matrices. With this condition, the absolute neutrino mass can be obtained in terms of the mass-squared differences. When combined with the accumulated experimental data this condition predicts two types of mass hierarchies, with one of them characterized by $m_{ν_3} \approx -2m_{ν_1} \approx -2 m_{ν_2} \approx 0.063$ eV, and the other by $m_{ν_1} \approx -m_{ν_2} \approx 0.054$ eV and $m_{ν_3} \approx 0.0064$ eV. The mass ranges predicted is just below the cosmological upper bound of 0.23 eV from recent WMAP data and can be probed in the near future. We also point out some implications for direct laboratory measurement of neutrino masses, and the neutrino mass matrix.