Neutrinoless double-$\beta$ decay of ${}^{124}\mathrm{Sn}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$ in the Hamiltonian-based generator-coordinate method

We present a generator-coordinate method for realistic shell-model Hamiltonians that closely approximates the full shell model calculations of the matrix elements for the neutrinoless double-$\beta$ decay of ${}^{124}\mathrm{Sn}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$. We treat not only quadrupole deformations but also the proton-neutron pairing amplitudes as generator coordinates. We validate this method by calculating and comparing spectroscopic quantities with the exact shell model results and experimental data. Our Hamiltonian-based generator-coordinate method produces $0\nu \beta \beta$ matrix elements much closer to the shell model ones, compared to the existing energy-density-functional-based generator-coordinate approaches. The remaining overestimation of $0\nu \beta \beta$ nuclear matrix element suggests that additional correlations may be needed to be taken into account for ${}^{124}\mathrm{Sn}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$ when calculating with the Hamiltonian-based generator-coordinate method. The validation of this method may open the possibility of calculating $0\nu \beta \beta$ matrix element of ${}^{150}\mathrm{Nd}$ in a large shell-model space.

Figure 1

The calculated low-lying energy levels for ${}^{124}\mathrm{Sn}$, ${}^{124}\mathrm{Te}$, ${}^{130}\mathrm{Te}$, ${}^{130}\mathrm{Xe}$, ${}^{136}\mathrm{Xe}$, and ${}^{136}\mathrm{Ba}$, compared to the exact solutions of SM [9, 10] and experimental data [43].

Figure 2

The calculated vacancies of valence neutron orbitals for ${}^{124}\mathrm{Sn}$, ${}^{124}\mathrm{Te}$, and ${}^{136}\mathrm{Ba}$, as well as the calculated occupancies of valence proton orbitals for ${}^{124}\mathrm{Te}$ ${}^{136}\mathrm{Xe}$, and ${}^{136}\mathrm{Ba}$.

Figure 3

Calculated $0\nu \beta \beta$ NMEs, compared to the values given by SM (denoted by “ISM-CMU”) [9, 10], EDF-based GCM employed non-relativistic Gogny D1S force (denoted by “NREDF”) [29] and relativistic PC-PK1 force (denoted by “REDF”) [30].

Figure 4

$I$-pair decomposition: contributions to the Gamow-Teller matrix elements for the $0\nu \beta \beta$ decay of ${}^{124}\mathrm{Sn}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$ from the configurations when two initial neutrons and two final protons have a certain total spin $I$, compared to SM [9, 10] calculations. CD-Bonn SRC parametrization was used.

Figure 5

The differences of Gamow-Teller part of NMEs between our GCM and SM calculations against the pair-spin $I$ for ${}^{124}\mathrm{Sn}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$.