  We consider Lifshitz-type scalar field theories that exhibit anisotropic
scaling laws near the ultraviolet fixed point, with explicit breaking of
Lorentz symmetry. It is shown that, when all momentum dependent vertex
operators are discarded, actions with anisotropy parameter z=3 in 3+1
dimensions generate Lorentz symmetry violating quantum corrections that are
suppressed by inverse powers of the momentum, so that the symmetry is sensibly
restored in the infrared region. In the ultraviolet region, the singular
behavior of the corrections is strongly smoothened: only logarithmic
divergences show up, producing very small changes of the couplings over a range
of momentum of many orders of magnitude. In the particular case where all
couplings are equal, the theory shows a Liouville-like potential and quantum
corrections are exactly summable, giving an asymptotically free theory.
However, the observed weakening of the divergences is not sufficient to avoid a
residual fine tuning of the mass parameter at a very high energy scale, in
order to recover a physically acceptable mass in the infrared region.
