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

Abstract

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) for the $0\nu \beta \beta$ decay exhibit large uncertainties depending on the models employed. This problem has affected the development of neutrino physics for many years. We have recently performed 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 the 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, ${}^{130}$Te, and ${}^{76}$Ge obtained by several groups and show that the uncertainty of the $0\nu \beta \beta$ NME is dramatically reduced. Based on this result, we calculate the effective neutrino mass, consistent with the current experimental lower limit of the half-life for the $0\nu \beta \beta$ decay. The results indicate that the extracted value of the effective neutrino mass does not yet reach the inverted-mass-hierarchy region allowed by neutrino oscillation data when the lightest neutrino mass is assumed to be below 10 meV. We also calculate the perturbed $0\nu \beta \beta$ and $2\nu \beta \beta$ NMEs of ${}^{110}$Pd.

19 pages, 14 figures