Nuclear Structure Aspects of Neutrinoless Double-$\beta$ Decay

We decompose the neutrinoless double-$\beta$ decay matrix elements into sums of products over the intermediate nucleus with two less nucleons. We find that the sum is dominated by the $J^{\pi }=0^{+}$ ground state of this intermediate nucleus for both the light and heavy neutrino decay processes. This provides a new theoretical tool for comparing and improving nuclear structure models. It also provides the connection to two-nucleon transfer experiments.

Figure 1

Nuclei involved in the calculations for the double-$\beta$ decay of ${}^{76}\mathrm{Ge}$.

Figure 2

Results for ${}^{76}\mathrm{Ge}$. The left-hand column shows the light-neutrino results ($\nu$) for the sum of the GT, $F$, and $T$ contributions. The middle column shows the light-neutrino results for the GT contribution only. The right-hand column shows the heavy-neutrino results ($N$) for the sum of the GT, $F$, and $T$ contribution. The bottom row shows the running sums for the $0^{+}$ intermediate states. The middle row shows the running sums for the $0^{+}$ and $2^{+}$ intermediate states. The top row shows the running sums for all intermediate states. The red dots are the exact results for the sum over all intermediate states.

Figure 3

Results for ${}^{48}\mathrm{Ca}$. See caption for Fig. 2.

Figure 4

Results for ${}^{82}\mathrm{Se}$. See caption for Fig. 2.

Figure 5

Schematic diagram for the configurations involved in double-$\beta$ decay. The top row shows the configuration changes for the orbitals near the Fermi surface for double-$\beta$ decay of a nucleus with mass $A$ through intermediate states of the nucleus with mass ($A-2$). The bottom row shows examples of important configuration admixtures to those above them in the top row. Proton occupations are shown on the left-hand side and neutron occupations are shown on the right hand side of each configuration. Crosses indicate particles and circles indicate holes.