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

TL;DR

This paper demonstrates that adding quadratic curvature terms to Einstein gravity does not alter graviton scattering amplitudes or the Newtonian potential, indicating no classical or quantum corrections from these terms.

Contribution

It proves that $R^2$ and $R^{ u ho} R_{ u ho}$ interactions do not affect graviton amplitudes or the Newtonian potential in effective field theory.

Findings

Graviton amplitudes remain unchanged with quadratic curvature terms.

No classical or quantum corrections to Newton's potential from $R^2$ terms.

Field redefinitions and unitarity imply the invariance of the potential.

Abstract

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.