Nuclear matrix element of $2νββ$ 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-$β$ ($2νββ$) 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 $\rightarrow$ $^{76}$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-$β$ 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 $2νββ$ 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 $2νββ$ NME is negligible. Besides, the 2BC in the transition operator is found to contribute $\sim 10\%$ quenching to the $2νββ$-decay NME of $^{76}$Ge.