Neutrinoless double-β decay of 124 Sn, 130 Te, and 136 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-$\\ensuremath{\\beta}$ 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 $0\\ensuremath{\\nu}\\ensuremath{\\beta}\\ensuremath{\\beta}$ matrix elements much closer to the shell model ones, compared to the existing energy-density-functional-based generator-coordinate approaches. The remaining overestimation of $0\\ensuremath{\\nu}\\ensuremath{\\beta}\\ensuremath{\\beta}$ 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 $0\\ensuremath{\\nu}\\ensuremath{\\beta}\\ensuremath{\\beta}$ matrix element of $^{150}\\mathrm{Nd}$ in a large shell-model space.