Long Range Forces in Quantum Gravity

TL;DR

The paper investigates leading quantum and semi-classical corrections to the Newtonian potential between two static masses in quantum gravity, arguing that long-range effects originate from massless propagators and can be captured within an effective field theory framework. Using Modanese's Euclidean functional-integral approach, they isolate the static (3-D) and ultrarelativistic (2-D) effective theories and compute the potential to order $G^2$, obtaining a classical post-Newtonian correction and a quantum correction with a definite sign: $V(R) = -{G M_1 M_2 \over R} \left[ 1 - {G(M_1+M_2) \over 2 R c^2} + {17 G \hbar \over 20 \pi R^2 c^3} \right]$. They contrast their results with Donoghue’s calculation, discuss the role of the infrared sector, and connect the static and high-energy scattering pictures to string perturbation theory, highlighting how low-energy field theory captures the dominant long-range behavior while string theory provides a UV-complete description. The work clarifies the infrared structure of quantum gravitational corrections and underscores the relevance of massless modes, with implications for the interface between quantum gravity and string theory in the low-energy regime.

Abstract

We calculate the leading quantum and semi-classical corrections to the Newtonian potential energy of two widely separated static masses. In this large-distance, static limit, the quantum behaviour of the sources does not contribute to the quantum corrections of the potential. These arise exclusively from the propagation of massless degrees of freedom. Our one-loop result is based on Modanese's formulation and is in disagreement with Donoghue's recent calculation. Also, we compare and contrast the structural similarities of our approach to scattering at ultra-high energy and large impact parameter. We connect our approach to results from string perturbation theory.