Two-neutrino $ββ$ decay to excited states at next-to-leading order
We study two-neutrino double-beta decay ($2νββ$) into first-excited $0^+_2$ states of nuclei used in $ββ$ decay experiments, including $^{76}$Ge, $^{82}$Se, $^{130}$Te, and $^{136}$Xe. We calculate the corresponding nuclear matrix elements (NMEs) within the nuclear shell model, using various Hamiltonians that describe well the spectroscopy of the initial and final nuclei. We evaluate the next-to-leading order (NLO) long-range NMEs recently introduced within chiral effective field theory, keeping three terms in the expansion of the energy denominator. In most cases, NLO contributions to the half-life are below 5%, but they can significantly increase due to cancellations in the leading-order Gamow-Teller NME. A detailed analysis in terms of nuclear deformation, including triaxiality, indicates that larger deformation differences between the initial and final states generally lead to smaller NMEs, but the seniority structure of the states also plays a relevant role. The lower range of our predicted half-lives, with uncertainties dominated by the nuclear Hamiltonian used, are slightly longer than the current experimental limit in $^{76}$Ge and consistent with the very recent half-life indication in $^{82}$Se.