  We study the renormalization of some dimension-4, 7 and 10 operators in a
class of nonlinear scalar-tensor theories. These theories are invariant under:
(a) linear diffeomorphisms which represent an exact symmetry of the full
non-linear action, and (b) global field-space Galilean transformations of the
scalar field. The Lagrangian contains a set of non-topological interaction
terms of the above-mentioned dimensionality, which we show are not renormalized
at any order in perturbation theory. We also discuss the renormalization of
other operators, that may be generated by loops and/or receive
loop-corrections, and identify the regime in which they are sub-leading with
respect to the operators that do not get renormalized. Interestingly, such
scalar-tensor theories emerge in a certain high-energy limit of the ghost-free
theory of massive gravity. One can use the non-renormalization properties of
the high-energy limit to estimate the magnitude of quantum corrections in the
full theory. We show that the quantum corrections to the three free parameters
of the model, one of them being the graviton mass, are strongly suppressed. In
particular, we show that having an arbitrarily small graviton mass is
technically natural.
