Neutrinoless double-$\beta$ decay matrix elements in large shell-model spaces with the generator-coordinate method

We use the generator-coordinate method (GCM) with realistic shell-model interactions to closely approximate full shell-model calculations of the matrix elements for the neutrinoless double-$\beta$ decay of ${}^{48}\mathrm{Ca}$, ${}^{76}\mathrm{Ge}$, and ${}^{82}\mathrm{Se}$. We work in one major shell for the first isotope, in the $f_{5/2}p{g}_{9/2}$ space for the second and third, and finally in two major shells for all three. Our coordinates include not only the usual axial deformation parameter $\beta$, but also the triaxiality angle $\gamma$ and neutron-proton pairing amplitudes. In the smaller model spaces our matrix elements agree well with those of full shell-model diagonalization, suggesting that our Hamiltonian-based GCM captures most of the important valence-space correlations. In two major shells, where exact diagonalization is not currently possible, our matrix elements are only slightly different from those in a single shell.

Figure 1

GCM results for the Gamow-Teller part of $0\nu \beta \beta$ matrix elements of ${}^{48}\mathrm{Ca}$, ${}^{54}\mathrm{Ti}$, and ${}^{54}\mathrm{Cr}$, compared with the results of exact diagonalization.

Figure 2

Projected potential-energy surfaces produced by the GCN2850 interaction, with the isoscalar-pairing amplitude $\varphi =0$, in the ($\beta ,\gamma$) plane for ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$.

Figure 3

Low-lying excitation spectra of ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$ produced in one shell by the GCM with the GCN2850 interaction, with and without triaxial deformation (labeled by the parameter $\gamma$). The results from the exact diagonalization of the shell-model Hamiltonian appear for comparison [41].

Figure 4

Occupancies of valence neutron and proton orbits produced by the interaction $pfsdg$ (see text) for ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$, following the adjustment of the single-particle energies for levels in the lower shell. The measured occupancies are from Refs. [52, 53].

Figure 5

Potential-energy surfaces for ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$ with the Hamiltonian $pfsdg$.

Figure 6

Calculated low-lying excitation spectra of ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$ produced by the Hamiltonian $pfsdg$ alongside the exact result with GCN2850 in one shell and experimental data [54].

Figure 7

GCM matrix elements $M^{0\nu }$ compared with those of the shell-model (SM), with either the JUN45 [35], CN2850 [6] KB3G [16], or SDPFMU-DB [4] interactions. The term $pfsdg$ denotes the two-shell interaction used here for $A\approx 80$ nuclei.