  For the first time, half-lives and energy spectra of forbidden $\beta$ decays
are calculated within the realistic shell model. Namely, we approach this issue
starting from a realistic nucleon-nucleon potential and deriving effective
Hamiltonians and decay operators. Our goal is to explore the sensitivity of the
shape of calculated energy spectra to the renormalization of forbidden $\beta$
-decay operators, an operation that allows to take into account those
configurations that are not explicitly included in the chosen model space. The
region that has been considered for this investigation are nuclei outside the
$^{78}$Ni core, more precisely we have studied the second-forbidden $\beta$
decays of $^{94}$Nb and $^{99}$Tc, and fourth-forbidden $\beta$ decays of
$^{113}$Cd and $^{115}$In, that are currently of a renewed experimental
interest in terms of novel spectroscopic techniques. Our results evidence that
the introduction of a renormalized $\beta$-decay operator leads to a marked
improvement of the reproduction of experimental half-lives. As regards the
spectra of both second-forbidden and fourth-forbidden decays, we have found
that their calculated shapes are in good agreement with the observed ones, even
if scarcely responsive to the renormalization of the decay operator. We carry
out also a detailed inspection of the different components of the calculated
spectra for a deeper insight about their role in reproducing the experimental
shapes.
