attempted definitions that have been suggested in the literature [@runningG].

Moreover there is an extensive approach to quantum gravity that relies on the running of the gravitational coupling. Within the hypothesis of Asymptotic Safety [@Weinberg:1980gg; @asymsafety], a suitably normalized version of the gravitational coupling is proposed to run to an ultraviolet fixed point. In Euclidean space, one treats the dimensionless combination $g= Gk_E^2$, where $k_E$ is a measure of the Euclidean energy, and the fixed point is described by $$g= Gk_E^2 \to g_*   ~~~~{\rm as} ~~~k_E \to \infty\,.$$ This would imply that the running gravitational strength itself vanishes at large energy, $G(k_E)\to 0$. This behavior is often summarized by a function $$G(k_E) = \frac{G_N}{1+\alpha G_N k_E^2}\,,
\label{runningform}$$ where $G_N$ is the gravitational coupling at zero energy.

Gravitational corrections are calculable at low energy using effective field theory [@Donoghue:1994dn]. However the effective field theory presents a very different picture of quantum effects - one in which the gravitational constant does [*not*]{} run. Because of the dimensional coupling, loops do not lead to a renormalization of the leading gravitational action $R$, i.e. the one that contains the gravitational constant $G$, but rather renormalize higher order terms in the action that come with 