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

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