Neutrinoless double-$\beta$ decay of ${}^{82}\mathrm{Se}$ in the shell model: Beyond the closure approximation

We recently proposed a method [R. A. Senkov and M. Horoi, Phys. Rev. C 88, 064312 (2013)] to calculate the standard nuclear matrix elements for neutrinoless double-$\beta$ decay ($0\nu \beta \beta$) of ${}^{48}\mathrm{Ca}$ going beyond the closure approximation. Here we extend this analysis to the important case of ${}^{82}\mathrm{Se}$, which was chosen as the base isotope for the upcoming SuperNEMO experiment. We demonstrate that by using a mixed method that considers information from closure and nonclosure approaches, one can get excellent convergence properties for the nuclear matrix elements, which allows one to avoid unmanageable computational costs. We show that in contrast with the closure approximation the mixed approach has a very weak dependence on the average closure energy. The matrix elements for the heavy neutrino-exchange mechanism that could contribute to the $0\nu \beta \beta$ decay of ${}^{82}\mathrm{Se}$ are also presented.

Figure 1

$J_{\kappa }$ decomposition: contributions of the intermediate states $|\kappa \rangle$ with certain spin and parity $J^{\pi }$ to the running nonclosure Gamow-Teller (solid colors) and Fermi (dashed colors) matrix elements for the $0\nu \beta \beta$ decay of ${}^{82}\mathrm{Se}$ (light neutrino exchange). Solid black and dashed white bars correspond to the positive-parity states, while solid gray and shaded black bars represent the states with negative parity. CD-Bonn SRC parametrization was used.

Figure 2

$I$ decomposition: contributions to the running nonclosure Gamow-Teller and Fermi matrix elements for the $0\nu \beta \beta$ decay of ${}^{82}\mathrm{Se}$ (light neutrino exchange) from the configurations when two initial neutrons $|24\rangle$ (and two final protons $|13\rangle$) have certain total spin $I$, $\langle 13,I\left|{\mathcal{O}}^{\alpha }\right|24,I\rangle$. The grayscale pattern and the SRC parametrization scheme are the same as in Fig. 1.

Figure 3

Convergence of NME for the $0\nu \beta \beta$ decay of ${}^{82}\mathrm{Se}$ (light neutrino exchange) calculated within different approximations: mixed (solid curve), running nonclosure (dashed curve), pure closure (dotted curve, does not depend on $N$), and running closure (dash-dotted curve). All calculations were done with CD-Bonn SRC and $\langle E\rangle =10$ MeV.

Figure 4

Dependence of mixed NME (light neutrino exchange) on the cutoff parameter $N$ calculated for different average closure energies $\langle E\rangle$. The main panel: $\langle E\rangle =1$ MeV (solid curve), $\langle E\rangle =3.4$ MeV (dash-dotted curve), $\langle E\rangle =7$ MeV (dashed curve), and $\langle E\rangle =1$ MeV (dotted curve). The insert shows the uncertainty in the value of mixed NME corresponding to the shaded area from the main panel.

Figure 5

Dependence of mixed and pure closure total NME for the $0\nu \beta \beta$ decay of ${}^{82}\mathrm{Se}$ (light neutrino exchange) on the average closure energy $\langle E\rangle$. Matrix elements presented as: mixed with CD-Bonn SRC (solid curve), closure with CD-Bonn SRC (dash-dotted), mixed with AV18 SRC (dashed curve), and closure with AV18 (dotted curve).

Figure 6

$J_{\kappa }$ decomposition: contributions of the intermediate states $|\kappa \rangle$ with certain spin and parity $J^{\pi }$ to the Gamow-Teller (solid colors) and Fermi (dashed colors) matrix elements for the $0\nu \beta \beta$ decay of ${}^{82}\mathrm{Se}$ (heavy neutrino exchange). Solid black and dashed white bars correspond to the positive-parity states, while solid gray and dashed black bars represent the states with negative parity. All calculations were done with CD-Bonn SRC.