Two-neutrino double-$\beta$ decay matrix elements based on a relativistic nuclear energy density functional

Nuclear matrix elements (NMEs) for two-neutrino double-$\beta$ decay ($2\nu \beta \beta$) are studied in the framework of a relativistic nuclear energy density functional. The properties of nuclei involved in the decay are obtained using the relativistic Hartree-Bardeen-Cooper-Schrieffer theory and relevant nuclear transitions are described using the relativistic proton-neutron quasiparticle random phase approximation based on the relativistic energy density functional. Three effective interactions are employed, including density-dependent meson-exchange (DD-ME2) and point-coupling interactions (DD-PC1 and DD-PCX), and pairing correlations are described consistently both in $T=1$ and $T=0$ channels using a separable pairing interaction. The optimal values of $T=0$ pairing strength parameter $V_{0pp}$ are constrained by the experimental data on $\beta$-decay half-lives. The $2\nu \beta \beta$ matrix elements and half-lives are calculated for several nuclides experimentally known to undergo this kind of decay:${}^{48}\mathrm{Ca}, {}^{76}\mathrm{Ge}, {}^{82}\mathrm{Se}, {}^{96}\mathrm{Zr}, {}^{100}\mathrm{Mo}, {}^{116}\mathrm{Cd}, {}^{124}\mathrm{Xe}, {}^{128}\mathrm{Te}, {}^{130}\mathrm{Te}, {}^{136}\mathrm{Xe}$, and ${}^{150}\mathrm{Nd}$. The model dependence of the NMEs and their sensitivity on $V_{0pp}$ is investigated, and the NMEs obtained using optimal values of $V_{0pp}$ are discussed in comparison to previous studies. The results of the present work represent an important benchmark for the future applications of the relativistic framework in studies of neutrinoless double-$\beta$ decay.

Figure 1

The NMEs for $2\nu \beta \beta$ decay based on Gamow-Teller transitions, shown as a function of the maximal two-quasiparticle energy. The DD-ME2 interaction is used in calculations. Matrix elements are dimensionless.

Figure 2

The running sum of the GT NMEs for the $2\nu \beta \beta$ decay of ${}^{48}\mathrm{Ca}$ for the DD-ME2 interaction, shown as a function of maximal excitation energy ${\mathrm{E}}_{\mathrm{exc}}$ in the intermediate nucleus. The cases with and without $T=0$ pairing are shown separately.

Figure 3

The same as in Fig. 2, but for ${}^{76}\mathrm{Ge}$.

Figure 4

Running sums for the GT NMEs for DD-ME2 interaction and optimal values of $V_{0pp}$ from Table 2, for the $2\nu \beta \beta$ decays of ${}^{82}\mathrm{Se}$–${}^{150}\mathrm{Nd}$, shown as a function of maximal excitation energy $E_{\mathrm{exc}}$ in the intermediate nucleus.

Figure 5

The dependence of the GT NMEs for $2\nu \beta \beta$ decay on the isoscalar pairing strength $V_{0pp}$ for ${}^{48}\mathrm{Ca}$, using DD-ME2 interaction, in comparison to the result obtained from the experimental data on $2\nu \beta \beta$ decay [80].

Figure 6

The difference between the NMEs calculated using two prescriptions for the GT $2\nu \beta \beta$ decay matrix element, shown as a function of the isoscalar pairing strength $V_{0pp}$ (see text for details). “${\mathrm{M}}_{GT,\mathrm{simpl}.}$” refers to the usual prescription for the overlap factor in Eq. (7), while “${\mathrm{M}}_{GT,\mathrm{Simkovic}}$” refers to the prescription in Eq. (8).

Figure 7

The function $C\left(r_{12}\right)$, showing the contributions to the NME at nucleon separations $r_{12}$, for the Gamow-Teller $2\nu \beta \beta$ decay of ${}^{48}\mathrm{Ca}$, for $V_{0pp}$ from 0.0 to 1.5.

Figure 8

The function $C\left(r_{12}\right)$, showing the contributions to the NME at nucleon separations $r_{12}$, for the Gamow-Teller $2\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$–${}^{150}\mathrm{Nd}$, evaluated at optimal values of $V_{0pp}$.

Figure 9

The dependence of the NMEs for $2\nu \beta \beta$ decay on the isoscalar pairing strength $V_{0pp}$ for $48<A<150$ nuclei that decay through the $2\nu \beta \beta$ channel, in comparison to the result obtained from the experimental data [80].

Figure 10

The dependence of the NME for $2\nu \beta \beta$ decay on the isoscalar pairing strength $V_{0pp}$ for $A=48$–150 nuclides.

Figure 11

Comparison of the NMEs for ${}^{48}\mathrm{Ca}$ using the DD-PC1 and DD-PCX interactions and the value obtained from experimental data [127].

Figure 12

The dependence of the NMEs on the isoscalar pairing strength $V_{0pp}$ for the $2\nu \beta \beta$ decay for the set of nuclides in the mass range ${}^{76}\mathrm{Ge}$–${}^{150}\mathrm{Nd}$, for all relativistic interactions employed in this work shown in comparison to the values obtained from the experimental data [127].

Figure 13

Summary of the $2\nu \beta \beta$ decay NMEs from the REDF-QRPA with DD-ME2, DD-PC1 and DD-PCX interactions with optimal $T=0$ pairing together with respective $1\sigma$ uncertainties, compared to the calculations based on the pn-QRPA by Deppisch and Suhonen [124], Pirinen and Suhonen [1], and Šimkovic et al. [84], IBM [120], ISM [112], and the ET outlined in Ref. [126]. The NMEs from the experimental data [80] are also shown.

Figure 14

The NMEs for the DD-ME2 interaction, calculated with the $Q$ values derived self-consistently in the RH-BCS framework, shown next to the NMEs for the same interaction, but with the $Q$ values based on experimental data [130].