Determination of neutrino mass ordering in future $^{76}$Ge-based neutrinoless double-beta decay experiments

Motivated by recent intensive experimental efforts on searching for neutrinoless double-beta decays, we perform a detailed analysis of the physics potential of the experiments based on $^{76}\mathrm{Ge}$. Assuming no signals, current and future experiments could place a $90\%$ lower limit on the half life $T^{0\nu}_{1/2} \gtrsim 4\times 10^{26}~{\rm yr}$ and $T^{0\nu}_{1/2} \gtrsim 7\times 10^{27}~{\rm yr}$, respectively. Then, how to report an evidence for neutrinoless double-beta decays is addressed by following the Bayesian statistical approach. For the first time, we present a quantitative description of experimental power to distinguish between normal and inverted neutrino mass orderings. Taking an exposure of $10^{4}~{\rm kg}\cdot{\rm yr}$ and a background rate of $10^{-4}~{\rm counts}/({\rm keV}\cdot{\rm kg}\cdot{\rm yr})$, we find that a moderate evidence for normal neutrino mass ordering (i.e., with a Bayes factor ${\cal B}$ given by $\ln({\cal B}) \simeq 2.5$ or a probability about $92.3\%$ according to the Jeffreys scale) can be achieved if the true value of effective neutrino mass $m^{}_{\beta\beta}$ turns out to be below $0.01~{\rm eV}$.