Nuclear matrix element of $2\nu \beta \beta$ decay of ${}^{76}$Ge: roles of high-lying states and two-body currents

Abstract

We present a microscopic analysis of the nuclear matrix element (NME) of the two-neutrino double-$\beta$ ($2\nu \beta \beta$) decay for open-shell heavy deformed nuclei, taking into account the fact that nuclear level density increases rapidly with excitation energy as well as the contribution of two-body current (2BC). Taking ${}^{76}$Ge $\to$ ${}^{76}$Se decay as an example, we found that due to the rapid increase of the level density of the intermediate nucleus ${}^{76}$As with excitation energy $E_n$, the single-$\beta$ Gamow-Teller (GT) matrix elements become highly fragmented with very small magnitude, and exhibit seemingly random sign patterns at high $E_n$ region. This leads to an effective cancellation at high $E_n$ region in calculating the $2\nu \beta \beta$ NME which then turns out to converge at $E_n\lesssim 5$ MeV, indicating that the contribution of high-lying states of the intermediate nucleus to $2\nu \beta \beta$ NME is negligible. Besides, the 2BC in the transition operator is found to contribute $\sim 10\%$ quenching to the $2\nu \beta \beta$-decay NME of ${}^{76}$Ge.