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abstract: 'We show that there is no useful and universal definition of a running gravitational constant, G(E), in the perturbative regime below the Planck scale. By consideration of the loop corrections to several physical processes, we show that the quantum corrections vary greatly, in both magnitude and sign, and do not exhibit the required properties of a running coupling constant. We comment on the potential challenges of these results for the Asymptotic Safety program.'
author:
- 'Mohamed M. Anber${}^{1}$'
- 'John F. Donoghue${}^{2,3}$'
title: '[**On the running of the gravitational constant** ]{}'
---

Introduction
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The concepts of running coupling constants and the renormalization group are of great utility in renormalizable field theories. It is tempting to attempt a definition of a running gravitational constant $G(q^2)$ in the context of quantum general relativity. As a simple example, one could take the quantum correction to the gravitational potential between heavy masses [@potential][^1] $$V(r) = -G\frac{Mm}{r}\left[1+ \frac{41}{10\pi}\frac{G}{r^2}\right]\,,$$ and turn it into a running coupling $$G(r) = G\left[1+ \frac{41}{10\pi}\frac{G}{r^2}\right]$$ which would imply that the gravitational strength gets stronger at short distance. We will return to the weaknesses of this particular definition in Sec. 8 below, but it is only one of many 