$0\nu \beta \beta$ nuclear matrix elements and the occupancy of individual orbits

The measured occupancies of valence orbits in ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$ are used as a guideline for modification of the effective mean field energies that results in better description of these quantities. With them, in combination with the self-consistent renormalized quasiparticle random phase approximation (SRQRPA) method that ensures conservation of the mean particle number in the correlated ground state, we show that the resulting $0\nu \beta \beta$ nuclear matrix element for the ${}^{76}\mathrm{Ge}\to {}^{76}\mathrm{Se}$ transition is reduced by $~25\%$ compared to the previous QRPA value, and therefore the difference between the present approach and the interacting shell model predictions becomes correspondingly smaller. Analogous modification of the mean field energies for the $A=82$ system also results in a reduction of $0\nu \beta \beta$ matrix element for the ${}^{82}\mathrm{Se}\to {}^{82}\mathrm{Kr}$ transition, making it also closer to the shell model prediction.

Figure 1

Comparison of the neutron levels in the Woods-Saxon potential used in Ref. [3] (WS) and the adjusted mean field energies used here and described in the text. Symbols λ indicate the chemical potential and the crosses indicate the occupancy of the individual orbits.

Figure 2

Comparison of the proton levels of the Woods-Saxon potential used in [3] and the adjusted mean field energies used here and described in the text. Notation as in Fig. 1.

Figure 3

The running sum of $M^{2\nu }$, Eq. (12). The dashed line corresponds to Ref. [3], the full line is the present result, and the dot-and-dashed line is the experimental result [17]. In the insert the first 4 MeV of excitation energy are shown in detail.

Figure 4

The contribution of individual neutron orbit pairs to $M^{0\nu }$. The entries are normalized so that their sum is unity.

Figure 5

The contribution of individual proton orbit pairs to $M^{0\nu }$. The entries are normalized so that their sum is unity.

Figure 6

The contribution of the initial neutron orbits (combination $n,n^{\text{'}}$) (along the $y$ axis) plotted against the analogous combinations $p,p^{\text{'}}$ (along the $x$ axis) of the final proton orbits. The entries are again normalized to unity. For notation along the axes, see text.