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.