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

TL;DR

This paper introduces a generator-coordinate method based on realistic shell-model Hamiltonians that accurately approximates full shell model calculations for neutrinoless double-beta decay matrix elements in certain isotopes, improving upon previous approaches.

Contribution

The paper develops and validates a Hamiltonian-based generator-coordinate method that closely reproduces shell model results for neutrinoless double-beta decay matrix elements, incorporating quadrupole deformations and proton-neutron pairing.

Findings

Method produces matrix elements closer to shell model results than energy-density-functional approaches.

Validation shows good agreement with exact shell model calculations and experimental data.

Remaining overestimation indicates the need for additional correlations in the model.

Abstract

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 $^{124}$Sn, $^{130}$Te, and $^{136}$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 $^{124}$Sn, $^{130}$Te, and $^{136}$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 $^{150}$Nd in a large shell-model space.