  The investigation of UV divergences is a relevant step in better
understanding of a new theory. In this work the one-loop divergences in the
free field sector are obtained for the popular Galileons model. The
calculations are performed by the generalized Schwinger-DeWitt technique and
also by means of Feynman diagrams. The first method can be directly generalized
to curved space, but here we deal only with the flat-space limit. We show that
the UV completion of the theory includes the $\pi \Box^4\pi$ term. According to
our previous analysis in the case of quantum gravity, this means that the
theory can be modified to become superrenormalizable, but then its physical
spectrum includes two massive ghosts and one massive scalar with positive
kinetic energy. The effective approach in this theory can be perfectly
successful, exactly as in the higher derivative quantum gravity, and in this
case the non-renormalization theorem for Galileons remains valid in the
low-energy region.
