Accurate shell-model nuclear matrix elements for neutrinoless double-$\beta$ decay

We investigate a novel method of accurate calculation of the neutrinoless double-$\beta$ decay shell-model nuclear matrix elements for the experimentally relevant case of ${}^{76}\mathrm{Ge}$. We demonstrate that with the new method the nuclear matrix elements have perfect convergence properties and, using only the first 100 intermediate states of each spin, the matrix elements can be calculated with better than 1% accuracy. Based on the analysis of neutrinoless double-$\beta$ decays of ${}^{48}\mathrm{Ca},{}^{82}\mathrm{Se}$, and ${}^{76}\mathrm{Ge}$ isotopes, we propose a new method to estimate the optimal values of the average closure energies at which the closure approximation gives the most accurate nuclear matrix elements. We also analyze the nuclear matrix elements for the heavy-neutrino-exchange mechanism, and we show that our method can be used to quench contributions from different intermediate spin states.

Figure 1

Dependence of mixed NMEs (light-neutrino exchange) on the cutoff parameter $N$ calculated for different average closure energies $\langle E\rangle$. $\langle E\rangle =2$ MeV (solid curve), $\langle E\rangle =3.4$ MeV (dash-dotted curve), $\langle E\rangle =7$ MeV (dashed curve), and $\langle E\rangle =10$ MeV (dotted curve). Inset: Uncertainty in the value of mixed NMEs corresponding to the shaded area.

Figure 2

$J$ decomposition: contributions of the intermediate states $|\kappa \rangle$ with a certain spin and parity $J^{\pi }$ to the nonclosure Gamow-Teller (dark colors) and Fermi (light colors) matrix elements for the $0\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$ (light-neutrino exchange). Solid black and white bars correspond to positive-parity states, while shaded bars represent states with a negative parity. The CD-Bonn SRC parametrization was used.

Figure 3

Optimal closure energies $\langle E\rangle$ calculated for different isotopes and effective interactions. Fictitious $0\nu \beta \beta$decays: ${}^{44}\mathrm{Ca}$ (diamonds) and ${}^{46}\mathrm{Ca}$ (squares). Real decays: ${}^{48}\mathrm{Ca}$ (circles), ${}^{76}\mathrm{Ge}$ (upward triangle), and ${}^{82}\mathrm{Se}$ (downward triangle). Effective interactions considered are GXPF1A, FPD6, and KB3G for Ca and JUN45 for Ge and Se isotopes.