Quantum Gravitational Corrections to the Nonrelativistic Scattering Potential of Two Masses

TL;DR

This work treats general relativity as an effective field theory and computes the full nonanalytic part of the one-loop scattering amplitude for two masses to extract long-range corrections to the gravitational interaction. By defining the nonrelativistic potential from the scattering amplitude and Fourier transforming, it derives the complete potential incorporating both classical post-Newtonian and quantum corrections, giving $V(r) = -\frac{G m_1 m_2}{r}\left[1+3\frac{G(m_1+m_2)}{r}+\frac{41}{10\pi}\frac{G\hbar}{ r^2}\right]$, with the classical part agreeing up to coordinate choices and the quantum part presented as a definitive low-energy prediction. The paper also discusses ambiguities in the potential under coordinate transformations and argues for a well-defined quantum correction at this order within a specified Hamiltonian framework. These results provide a concrete, gauge-invariant link between EFT graviton dynamics and long-range quantum gravitational effects, reinforcing the view of gravity as a low-energy EFT with calculable quantum corrections, albeit extremely small in magnitude.

Abstract

We treat general relativity as an effective field theory, obtaining the full nonanalytic component of the scattering matrix potential to one-loop order. The lowest order vertex rules for the resulting effective field theory are presented and the one-loop diagrams which yield the leading nonrelativistic post-Newtonian and quantum corrections to the gravitational scattering amplitude to second order in G are calculated in detail. The Fourier transformed amplitudes yield a nonrelativistic potential and our result is discussed in relation to previous calculations. The definition of a potential is discussed as well and we show how the ambiguity of the potential under coordinate changes is resolved.

Figures (6)

• Figure 1: The tree diagram giving Newtons law.
• Figure 2: The box and crossed box diagrams which contribute to the non-analytic component of the potential.
• Figure 3: The set of triangule diagrams contributing to the scattering potential.
• Figure 4: The double-seagull diagram contribution to the scattering potential.
• Figure 5: The class of the graviton vertex corrections which yield nonanalytic corrections to the potential.