A note on the absence of $R^2$ corrections to Newton's potential

Understand

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.