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

TL;DR

The paper investigates whether quadratic curvature corrections to gravity, encoded by terms like $R^2$ and $R^{\mu\nu}R_{\mu\nu}$ in an effective field theory of gravity, modify the Newtonian potential. It combines three complementary approaches—field redefinitions via the S-matrix equivalence theorem, on-shell amplitude methods with factorisation and little-group constraints, and diagrammatic techniques—to show that all amplitudes involving two heavy scalars and any number of gravitons, and graviton-only amplitudes, are unaffected by these quadratic terms. The key results are that these corrections vanish at linear order and, using unitarity, yield no classical or quantum modifications to the Newtonian potential from $R^2$-type operators; the authors further argue that the cancellations extend to all orders in the quadratic couplings, albeit with caveats and guidance for future all-orders proofs. This work clarifies the role of quadratic curvature terms in EFT gravity and identifies the precise conditions under which they fail to alter long-range gravitational physics, while indicating that cubic corrections remain the leading potential source of new effects.

Abstract

We consider Einstein gravity with the addition of $R^2$ and $R^{μν} R_{μν}$ 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.

Figures (3)

• Figure 1: The one-loop unitarity cut in the $q^2$-channel contributing to the massive scalar scattering. Here the $\partial^4$ blob denotes the amplitude with one insertion of either $R^2$ or $R^{\mu \nu}R_{\mu \nu}$.
• Figure 2: The sum of the two diagrams contributing to the two-scalar two-graviton amplitudes to first order in $R^2$. The same result holds for $R^{\mu \nu} R_{\mu \nu}$. All external on-shell states are in $D$ dimensions, the legs labelled by Lorentz indices are off-shell. The relevant Feynman rules can be found in Appendix \ref{['sec::Feynmanrules']}.
• Figure 3: An example of higher-loop cut diagram contributing to the Newtonian potential. The $\partial^4$ symbol denotes an insertion of either $R^2$ or $R^{\mu \nu}R_{\mu \nu}$.