  We investigate some quantum features of the bumblebee electrodynamics in flat
spacetimes. The bumblebee field is a vector field that leads to a spontaneous
Lorentz symmetry breaking. For a smooth quadratic potential, the massless
excitation (Nambu-Goldstone boson) can be identified as the photon, transversal
to the vacuum expectation value of the bumblebee field. Besides, there is a
massive excitation associated with the longitudinal mode and whose presence
leads to instability in the spectrum of the theory. By using the
principal-value prescription, we show that no one-loop radiative corrections to
the mass term is generated. Moreover, the bumblebee self-energy is not
transverse, showing that the propagation of the longitudinal mode can not be
excluded from the effective theory.
