Two-neutrino double-$\beta$ decay in the mapped interacting boson model

A calculation of two-neutrino double-$\beta$ ($2\nu\beta\beta$) decay matrix elements within the interacting boson model (IBM) that is based on the nuclear density functional theory is presented. The constrained self-consistent mean-field (SCMF) calculation using a universal energy density functional (EDF) and a pairing interaction provides potential energy surfaces with triaxial quadrupole degrees of freedom for even-even nuclei corresponding to the initial and final states of the $2\nu\beta\beta$ decays of interest. The SCMF energy surface is then mapped onto the bosonic one, and this procedure determines the IBM Hamiltonian for the even-even nuclei. The same SCMF calculation provides the essential ingredients of the interacting boson fermion-fermion model (IBFFM) for the intermediate odd-odd nuclei and the Gamow-Teller and Fermi transition operators. The EDF-based IBM and IBFFM provide a simultaneous description of excitation spectra and electromagnetic transition rates for each nucleus, and single-$\beta$ and $\beta\beta$ decay properties. The calculated $2\nu\beta\beta$-decay nuclear matrix elements are compared with experiment and with those from earlier theoretical calculations.