The upper limit on the mass of the Majorana neutrino, extracted from the limits on the nonobservation of the neutrinoless double-$\beta$ ($0\nu \beta \beta$) decay, is hampered by uncertainties in the matrix elements of the transition operators. Recently, we have shown that the values of the effective axial-vector current coupling constants ($g_A^{\text{eff}}$) for the $0\nu \beta \beta$ and the two-neutrino double-$\beta$ decays are close. This striking result was obtained for the first time by including vertex corrections and two-body currents in these matrix elements. In this Letter, we calculate the half-life for the $0\nu \beta \beta$ decay ($T_{1/2}^{0\nu }$) of ${}^{136}\mathrm{Xe}$ using this closeness and show the convergence of the half-life with respect to the variation of the method to determine $g_A^{\text{eff}}$. The closeness of the $g_A^{\text{eff}}$ of the two decay modes plays a decisive role in predicting $T_{1/2}^{0\nu }$. The appropriate value of $g_A^{\text{eff}}$ depends on the assumptions made for the sectors of the nuclear structure and transition operators of the calculations within the perturbation scheme. The value $g_A^{\text{eff}}\approx 1$ is obtained when the ${\mathrm{SkM}}^{*}$ is used to describe the nuclear structure component, while a smaller value of $g_A^{\text{eff}}$ is obtained by applying a less realistic interaction like the SGII one.

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• Nuclear density functional theory
• Nuclear many-body theory
• Nuclear structure & decays