The $2\nu \beta \beta$ decay from the ground to the first excited $2^{+}$ state is considered for eight nuclei where experimental data are available. The transition amplitude is calculated with a projected spherical single-particle basis, a fully renormalized proton-neutron quasiparticle random-phase approximation (pnQRPA) for the Gamow-Teller (GT) dipole transition operator and a renormalized QRPA for charge conserving quadrupole operators. Also for the transition operator the first-order boson expansion expression is employed. Using the phase space integral corresponding to the transition $0^{+}\to 2^{+}$ and the GT transition amplitude, the process half-life is readily obtained. The single-${\beta }^{\mp }$ transition strengths are studied as function of the energies of the fully renormalized pnQRPA with the gauge symmetry restored. The single-$\beta$ transition operators are used to calculate the ${log}_{10}ft$ values for the electron capture of the intermediate odd-odd nucleus to the mother nucleus as well as for the ${\beta }^{-}$ transition to the daughter nucleus. For the final state in the daughter nucleus the $B\left(E2\right)$ values for the transition $2^{+}\to 0^{+}$ and the half-life of the electromagnetic decaying state are calculated. The Ikeda sum rule for the mother nucleus is satisfied. The mentioned results are compared with the corresponding available data and a reasonable agreement is shown. The gauge projection quenches the half-life in the case of ${}^{150}\mathrm{Nd}, {}^{116}\mathrm{Cd}$, and ${}^{100}\mathrm{Mo}$ and enhances it for the remaining considered nuclei. Keeping the same parameters for the model Hamiltonian the ground to ground double-$\beta$ transition is also treated and a good agreement with the existing data is obtained.

• Received 28 May 2022
• Accepted 14 September 2022

DOI:https://doi.org/10.1103/PhysRevC.106.044301