  In the standard seesaw mechanism, finite corrections to the neutrino mass
matrix arise from 1-loop self-energy diagrams mediated by a heavy neutrino. We
study in detail these corrections and demonstrate that they can be very
significant, exceeding in several cases the tree-level result. We consider the
normal and inverted hierarchy spectra for light neutrinos and compute the
finite corrections to the different elements of the neutrino mass matrix.
Special attention is paid to their dependence with the parameters of the seesaw
model. Among the cases in which the corrections can be large, we identify the
fine-tuned models considered previously in the literature, where a strong
cancellation between the different parameters is required to achieve
compatibility with the experimental data. As a particular example, we also
analyze how these corrections modify the tribimaximal mixing pattern and find
that the deviations may be sizable, in particular for $\theta_{13}$. Finally,
we emphasize that due to their large size, the finite corrections to neutrino
masses have to be taken into account if one wants to properly scan the
parameter space of seesaw models.
