Competition between allowed and first-forbidden $\beta$ decay in $r$-process waiting-point nuclei within a relativistic beyond-mean-field approach

We compute $\beta$-decay half-lives of isotonic nuclear chains located at neutron shell closures $N=50$, $82$, $126$ and $184$, which are of particular importance for the $r$-process nucleosynthesis, and study the role of first-forbidden transitions in a framework that includes complex nucleonic correlations beyond the quasiparticle random phase approximation. Such correlations are accounted for by coupling single nucleons to collective degrees of freedom (nuclear vibrations), and are found essential to reproduce available experimental $\beta$-decay rates and more precise many-body methods. We find that the nucleon-vibration correlations tend to decrease the probability of decay via first-forbidden transition near stability, as they enhance Gamow-Teller transitions at low energy. While in the lighter systems allowed transitions dominate, the decay of $N=126$ and $N=184$ nuclei is found to occur to a large extent via first-forbidden transitions, and in particular those induced by $1^-$ and $0^-$ operators. Overall the many-body method based on nucleon-vibration coupling provides an ideal framework for future large-scale calculations. Upcoming experimental measurements of $\beta$-decay rates in the $N=126$ region by radioactive-beam facilities will be crucial in order to validate the approach.