Systematic shell-model analysis of $2\nu\beta\beta$ decay of $^{76}$Ge and $^{96}$Zr to the ground and excited states of $^{76}$Se and $^{96}$Mo

In this work, we have studied the $2\nu\beta\beta$ decay of $^{76}$Ge and $^{96}$Zr isotopes utilizing large-scale shell-model calculations. The GWBXG effective interaction has been employed in the calculation of $2\nu\beta\beta$-decay nuclear matrix elements (NMEs). We have tested the effective interaction by comparing the predicted spectroscopic properties, such as energy spectra and transition probabilities, with the available experimental data. The variation of cumulative NMEs with respect to the $1^+$ state energies of the intermediate nucleus is also studied, corresponding to $0^+_{\rm g.s.}\rightarrow0^+_{\rm g.s.}$, $0^+_{\rm g.s.}\rightarrow0^+_{2}$, and $0^+_{\rm g.s.}\rightarrow2^+_{1}$ transitions between the parent and grand-daughter nuclei. The effective values of axial-vector coupling strength ($g_A^{\rm eff}$) are calculated using the predicted NMEs and experimental half-lives for $0^+_{\rm g.s.}\rightarrow0^+_{\rm g.s.}$ transitions. The extracted half-lives for $0^+_{\rm g.s.}\rightarrow0^+_{2}$, and $0^+_{\rm g.s.}\rightarrow2^+_{1}$ transitions using the shell-model predicted NMEs are consistent with the recent experimental data. The comparison of the shell-model predicted NMEs with previous NMEs available in the literature is discussed. Also, the computed branching ratios for the $2\nu\beta\beta$ decay of $^{76}$Ge and both the $2\nu\beta\beta$ and single-$\beta$ decay of $^{96}$Zr are reported corresponding to the calculated $g_A^{\rm eff}$ values.