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

A calculation of the spectroscopic properties, energy levels and electromagnetic transitions and moments, of the ten nuclei ${}^{128,130}$Te, ${}^{128,130}$I, ${}^{128,130}$Xe, ${}^{129,131}$I, ${}^{127,129}$Te within the framework of the interacting boson model (IBM-2) and its extensions (IBFM-2 and IBFFM-2) is presented. The wave functions so obtained are used to calculate single-$\beta$ and $2\nu \beta \beta$ matrix elements for ${}^{128,130}$Te $\to$ ${}^{128,130}$I $\to$ ${}^{128,130}$Xe ecay. Use of the effective value of the axial vector coupling constant $g_{A,\text{eff},\beta }$ extracted from single-$\beta$ produces results for $2\nu \beta \beta$ in agreement with experiment.