Shell model studies of the ${}^{130}\mathrm{Te}$ neutrinoless double-$\beta$ decay

Most uncertainties regarding the theoretical study of the neutrinoless double-$\beta$ decay are related to the accuracy of the nuclear matrix elements that appear in the expressions of the lifetimes. We calculate the nuclear matrix elements for the $0\nu \beta \beta$ decay of ${}^{130}\mathrm{Te}$ in a shell model approach, using a recently proposed effective Hamiltonian. To ensure the reliability of the results, we investigate this Hamiltonian by performing calculations of spectroscopic quantities and comparing them to the latest experimental data available, and we analyze the $2\nu \beta \beta$- and the $0\nu \beta \beta$-decay nuclear matrix elements of ${}^{136}\mathrm{Xe}$. Finally, we report new nuclear matrix elements for the ${}^{130}\mathrm{Te}$ considering the light neutrino exchange and heavy neutrino exchange mechanisms, alongside an overview of some recent values reported in the literature.

Figure 1

The calculated energy levels for ${}^{130}\mathrm{Te},{}^{130}\mathrm{Xe},{}^{136}\mathrm{Xe}$, and ${}^{136}\mathrm{Ba}$ (left columns) compared to the experimentally determined ones (right columns). All level, except those denoted with subscript $2 (0_2^{+},2_2^{+}$, and $6_2^{+})$, correspond to yrast states.

Figure 2

Theoretical and experimental [57] neutron shell vacancies for ${}^{128}\mathrm{Te}$ and ${}^{130}\mathrm{Te}$.

Figure 3

Theoretical and experimental [57] neutron shell vacancies for ${}^{130}\mathrm{Xe}$ and ${}^{132}\mathrm{Xe}$.

Figure 4

Theoretical proton shell occupancies for ${}^{128}\mathrm{Te},{}^{130}\mathrm{Te},{}^{130}\mathrm{Xe}$, and ${}^{132}\mathrm{Xe}$.

Figure 5

Calculated ${}^{130}\mathrm{Te}$ GT strengths (solid line) compared to the experimental ones (dotted line) [60]. The inset presents the calculated and the experimental GT running sum.

Figure 6

Calculated ${}^{136}\mathrm{Xe}$ GT strengths (solid line) compared to the experimental ones (dotted line) [61]. The inset presents the calculated and the experimental GT running sum.

Figure 7

$J_{\kappa }$ decomposition: contributions of the intermediate states $|\kappa \rangle$ with certain spin and parity $J^{\pi }$ to the nonclosure Gamow-Teller and Fermi matrix elements for the $0\nu \beta \beta$ decay of ${}^{136}\mathrm{Xe}$ (light neutrino exchange). CD-Bonn SRC parametrization was used.

Figure 8

$I$-pair decomposition: contributions to the running nonclosure Gamow-Teller and Fermi matrix elements for the $0\nu \beta \beta$ decay of ${}^{136}\mathrm{Xe}$ (light neutrino exchange) from the configurations when two initial neutrons and two final protons have a certain total spin $I$ and parity (positive or negative). CD-Bonn SRC parametrization was used.

Figure 9

Same as Fig. 7, here for ${}^{136}\mathrm{Xe}$ heavy neutrino.

Figure 10

Same as Fig. 8, here for ${}^{136}\mathrm{Xe}$ heavy neutrino.

Figure 11

Same as Fig. 8, here for ${}^{130}\mathrm{Te}$ light neutrino closure NME decomposition.

Figure 12

Same as Fig. 8, here for ${}^{130}\mathrm{Te}$ heavy neutrino closure NME decomposition.

Figure 13

Comparison of light neutrino exchange $0\nu \beta \beta$ NME obtained with different nuclear structure methods. Columns left to right correspond to down to up in the legend box.

Figure 14

Comparison of heavy neutrino exchange $0\nu \beta \beta$ NME obtained with different nuclear structure methods. Columns left to right correspond to down to up in the legend box.