Non-Oscillation Probes of the Neutrino Mass Hierarchy and Vanishing U_e3

One of the outstanding issues in neutrino physics is the experimental determination of the neutrino mass hierarchy: Is the order of the neutrino masses ``normal'' - m_1^2<m_2^2<m_3^2 - or is it inverted - m_3^2<m_1^2<m_2^2, with m_2^2-m_1^2 << m_2^2,m_1^2? While this issue can be resolved in next-generation long-baseline nu_{mu} to nu_e neutrino oscillation studies if |U_{e3}|^2 is large enough, a clear strategy on how to resolve it if |U_{e3}|^2 is sufficiently small is still lacking. We study the capability of non-oscillation probes of neutrino masses to determine the neutrino mass ordering. We concentrate on studies of m_{nu_e}, the kinematical neutrino mass to which precise studies of tritium beta-decay are sensitive, m_{ee}, the effective mass to which the rate for neutrinoless double-beta decay is sensitive if the neutrinos are Majorana fermions, and Sigma, the sum of the neutrino masses, to which cosmological probes of the energy budget of the Universe are sensitive. We find that combined measurements of m_{ee}, Sigma, and m_{nu_e} are capable of establishing the neutrino mass hierarchy if these measurements are precise enough and if one ``gets lucky.'' We quantify the previous sentence in detail by performing a numerical analysis of a large number of theoretical data sets, for different measured values of m_{nu_e}, m_{ee}, and Sigma, keeping in mind the ultimate sensitivity that can be reached by next (and next-to-next) generation experiments.