  Nuclear $\beta$-decay in magic nuclei is investigated, taking into account
the coupling between particle and collective vibrations,on top of
self-consistent random phase approximation calculations based on Skyrme density
functionals. The low-lying Gamow-Teller strength is shifted downwards and at
times becomes fragmented; as a consequence, the $\beta$-decay half-lives are
reduced due to the increase of the phase space available for the decay. In some
cases, this leads to a very good agreement between theoretical and experimental
lifetimes: this happens, in particular, in the case of the Skyrme force SkM*,
that can also reproduce the line shape of the high energy Gamow-Teller
resonance as it was previously shown.
