Predictions for Neutrinoless Double-Beta Decay in the 3+1 Sterile Neutrino Scenario

We present accurate predictions of the effective Majorana mass $|m_{\beta\beta}|$ in neutrinoless double-$\beta$ decay in the standard case of $3\nu$ mixing and in the case of 3+1 neutrino mixing indicated by the reactor, Gallium and LSND anomalies. We have taken into account the uncertainties of the neutrino mixing parameters determined by oscillation experiments. It is shown that the predictions for $|m_{\beta\beta}|$ in the cases of $3\nu$ and 3+1 mixing are quite different, in agreement with previous discussions in the literature, and that future measurements of neutrinoless double-$\beta$ decay and of the effective light neutrino mass in $\beta$ decay or the total mass of the three lightest neutrinos in cosmological experiments may distinguish the $3\nu$ and 3+1 cases if the mass ordering is determined by oscillation experiments. We also present a relatively simple method to determine the minimum value of $|m_{\beta\beta}|$ in the general case of $N$-neutrino mixing.