Evaluation of the theoretical nuclear matrix elements for $\beta \beta$ decay of ${}^{76}\mathrm{Ge}$

The nuclear matrix elements for two-neutrino double-$\beta$ and zero-neutrino double-$\beta$ decay of ${}^{76}\mathrm{Ge}$ are evaluated in terms of the configuration-interaction (CI), quasiparticle random-phase approximation (QRPA), and interacting boson model methods. We point out deficiencies in all of these models and suggest ways that some of them can be corrected. The final results are obtained from the CI method corrected for configuration admixtures involving orbitals outside of the CI configuration space by using results from QRPA, many-body-perturbation theory, and the connections to related observables. The CI two-neutrino matrix element is reduced due to the inclusion of spin-orbit partners and to many-body correlations connected with Gamow-Teller $\beta$ decay. The CI zero-neutrino matrix element for the heavy neutrino is enhanced due to particle-particle correlations that are connected with the odd-even oscillations in the nuclear masses. The CI zero-neutrino matrix element for the light neutrino contains both types of correlations that approximately cancel each other. The uncertainty from short-range correlations is also considered.

Figure 1

The $\beta \beta$ decay of ${}^{76}\mathrm{Ge}$. The top part shows the conversion of two neutrons (blue on the left) into two protons (red on the right) in two-neutrino double-$\beta \left(2\nu \beta \beta \right)$ decay. The bottom part shows the $0\nu \beta \beta$ decay where a virtual neutrino is exchanged.

Figure 2

Nuclear matrix elements for $2\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$. The top point in green is the experimental value [47]. The QRPA results are shown for $g_{pp}^{T=0}=0.673$ (red dots) and $g_{pp}^{T=0}=0.643$ (red crosses). The CI results are shown for the JUN45 (dot), jj44bpn (cross), and gcn28:50 (triangle) Hamiltonians.

Figure 3

The $0N$ NME for heavy-neutrino decay of ${}^{76}\mathrm{Ge}$. See caption for Fig. 2. The QRPA point with the triangle is from Ref. [29].

Figure 4

The $0\nu$ NME for the light-neutrino decay of ${}^{76}\mathrm{Ge}$. See captions for Figs. 2 and 3.