Shell-model calculation of neutrinoless double-$\beta$ decay of ${}^{76}\mathrm{Ge}$

In this article we present an extension of our recent Rapid Communication [Phys. Rev. C 90, 051301(R) (2014)] where we calculate the nuclear matrix elements for neutrinoless double-$\beta$ decay of ${}^{76}\mathrm{Ge}$. For the calculations we use a novel method that has perfect convergence properties and allows one to obtain the nonclosure nuclear matrix elements for ${}^{76}\mathrm{Ge}$ with a 1% accuracy. We present a new way to calculate the optimal closure energy; using this energy with the closure approximation provides the most accurate closure nuclear matrix elements. In addition, we present a new analysis of the heavy-neutrino-exchange nuclear matrix elements, and we compare occupation probabilities and Gamow-Teller strength with experimental data.

Figure 1

Theoretical (t) and experimental (x) neutron occupancies of the $p$ orbitals, $f_{5/2}$ orbital (f), and $g_{9/2}$ orbital (g) for ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$. Data are taken from Ref. [27].

Figure 2

Same as Fig. 1, for proton occupancies. Data are taken from Ref. [28].

Figure 3

Running sum of the Gamow-Teller strength in ${}^{76}\mathrm{Ge}$: the red line represents the calculated sum and the blue line is based on the high-resolution charge-exchange data [29].

Figure 4

Convergence of NMEs (light-neutrino exchange) as a function of the cutoff parameter $N$ calculated with different approximations: mixed (solid black curve), closure (solid red curve), running nonclosure (dashed black curve), and running closure (dashed blue curve). All calculations were done using the CD-Bonn SRC and $\langle E\rangle =9.41$ MeV [26].

Figure 5

Dependence of mixed and closure NMEs for the $0\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$ (light-neutrino exchange) on the average closure energy $\langle E\rangle$. MNEs: closure with the CD-Bonn SRC (dashed black curve), mixed with the CD-Bonn SRC (solid black curve), closure with the AV18 SRC (dashed red curve), and mixed with the AV18 SRC (solid red curve).

Figure 6

$I$ decomposition, light-neutrino exchange: contributions to the running nonclosure Gamow-Teller (vertically striped blue bars) and Fermi (horizontally striped black bars) matrix elements for the $0\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$ from the configurations when two initial neutrons $|24\rangle$ (and two final protons $|13\rangle$) have certain total spin $I,\langle 13,I\left|{\mathcal{O}}^{\alpha }\right|I,24\rangle$. Both parities are included and the CD-Bonn SRC parametrization was used.

Figure 7

$J$ decomposition, heavy-neutrino exchange: contributions of the intermediate states $|\kappa \rangle$ with certain spin and parity $J^{\pi }$ to the Gamow-Teller (blue and red bars) and Fermi (black and green bars) matrix elements for the $0\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$. Diagonally striped bars correspond to the contributions of positive-parity states, while horizontally and vertically striped bars represent states with a negative parity. All calculations were done with the CD-Bonn SRC.

Figure 8

$I$ decomposition: closure approximation Gamow-Teller and Fermi matrix elements (both parities) for the $0\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$, light-neutrino exchange. The calculation was performed with the optimal closure energy, $\langle E\rangle =3.5$ MeV. The results should be compared with the matrix elements presented in Fig. 6.

Figure 9

$I$ decomposition: closure approximation Gamow-Teller and Fermi matrix elements (both parities) for the $0\nu \beta \beta$ decay of ${}^{76}\mathrm{Ge}$, heavy-neutrino exchange.