Solution to the uncertainty problem of nuclear matrix element for neutrinoless double-$\beta$ decay

The neutrinoless double-$\beta$ decay ($0\nu\beta\beta$) of nuclei is one of the major research subjects of neutrino physics nowadays because of its influence on particle physics and astrophysics. The predicted nuclear matrix elements (NMEs) of the $0\nu\beta\beta$ decay have a large uncertainty depending on the models used to calculate them. This problem has affected the development of neutrino physics for many years. We have performed, recently, the calculation of the NMEs for the $0\nu\beta\beta$ and two-neutrino double-$\beta$ decay ($2\nu\beta\beta$) modes with a perturbed transition operator and found that the effective axial-vector current coupling ($g_A^\mathrm{eff}$) is similar for these two decay modes. Based on this finding, we calculate the $0\nu\beta\beta$ NMEs using the phenomenological $g_A^\mathrm{eff}$ that reproduces the measured half-life of the $2\nu\beta\beta$ decay. We apply this method to the NMEs for $^{136}$Xe obtained by several groups and show that the uncertainty of the $0\nu\beta\beta$ NME is dramatically reduced. Owing to this finding, we calculate the effective neutrino mass, consistent with the current experimental lower limit of the half-life for the $0\nu\beta\beta$ decay, and the results indicate that this effective neutrino mass value does not yet reach the inverted mass hierarchy region allowed by the neutrino oscillation data and the lightest neutrino mass assumed to be smaller than 10 meV.