  The possibility of removing the one-loop perturbative effects of lattice
artifacts by a proper choice of the lattice action is explored, and found to
depend crucially on the properties of the physical quantity considered. In this
respect the finite-space-volume mass gap m(L) is an improved observable. We
find an explicit momentum space representation of the one-loop contribution to
m(L) for arbitrary lattice actions in the case of two-dimensional O(N) sigma
models. We define a ``tree perfect'' Symanzik action and find that it formally
removes all one-loop lattice artifacts in m(L). On-shell improved actions do
not share this property.
