Neutrino masses with a “zero sum” condition:mν1+mν2+mν3=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 Ansätze to restrict the form of the neutrino mass matrix further. Here we propose that $m_{{\nu }_1}{+m}_{{\nu }_2}{+m}_{{\nu }_3}=0.\mathrm{}$ 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 $m_{{\nu }_3}\approx -{2m}_{{\nu }_1}\approx -{2m}_{{\nu }_2}\approx 0.063\mathrm{eV},$ and the other by $m_{{\nu }_1}\approx -m_{{\nu }_2}\approx 0.054\mathrm{eV}$ and $m_{{\nu }_3}\approx 0.0064\mathrm{eV}.$ The range predicted for $|m_{{\nu }_1}|+|m_{{\nu }_2}|+|m_{{\nu }_3}|$ is below the cosmological upper bound of 0.69 eV from recent Wilkinson Microwave Anisotropy Probe 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.