Estimation of nuclear matrix elements of double-$\beta$ decay from shell model and quasiparticle random-phase approximation

TL;DR

This paper develops a new method to reconcile and improve nuclear matrix element calculations for neutrinoless double-beta decay by combining shell model and QRPA approaches, enhancing reliability of neutrino mass estimates.

Contribution

It introduces a phenomenological modification technique to align shell model and QRPA NME calculations, addressing their individual limitations.

Findings

Modified NME values for $^{48}$Ca are consistent across methods.

Extrapolated shell model results toward larger valence spaces.

Improved agreement between shell model and QRPA calculations.

Abstract

The nuclear matrix element (NME) of the neutrinoless double-$\beta$ ($0\nu\beta\beta$) decay is an essential input for determining the neutrino effective mass, if the half-life of this decay is measured. The reliable calculation of this NME has been a long-standing problem because of the diversity of the predicted values of the NME depending on the calculation method. In this paper, we focus on the shell model and the QRPA. The shell model have a rich amount of the many-particle many-hole correlations, and the QRPA can obtain the convergence of the result of calculation with respect to the extension of the single-particle space. It is difficult for the shell model to obtain the convergence of the $0\nu\beta\beta$ NME with respect to the valence single-particle space. The many-body correlations of the QRPA are insufficient depending on nuclei. We propose a new method to modify phenomenologically the results of the shell model and the QRPA compensating the insufficient point of each method by using the information of other method complementarily. Extrapolations of the components of the $0\nu\beta\beta$ NME of the shell model are made toward a very large valence single-particle space. We introduce a modification factor to the components of the $0\nu\beta\beta$ NME of the QRPA. Our modification method gives similar values of the $0\nu\beta\beta$ NME of the two methods for $^{48}$Ca. The NME of the two-neutrino double-$\beta$ decay is also modified in a similar but simpler manner, and the consistency of the two methods is improved.