  We consider Einstein gravity with the addition of $R^2$ and $R^{\mu \nu}
R_{\mu \nu}$ interactions in the context of effective field theory, and the
corresponding scattering amplitudes of gravitons and minimally-coupled heavy
scalars. First, we recover the known fact that graviton amplitudes are the same
as in Einstein gravity. Then we show that all amplitudes with two heavy scalars
and an arbitrary number of gravitons are also not affected by these
interactions. We prove this by direct computations, using field redefinitions
known from earlier applications in string theory, and with a combination of
factorisation and power-counting arguments. Combined with unitarity, these
results imply that, in an effective field theory approach, the Newtonian
potential receives neither classical nor quantum corrections from terms
quadratic in the curvature.
