parameter $\mu$ does not enter the coordinate space potential at finite $r$ because the Fourier transform of a constant is a delta function. We keep $\mu \sim M_P$ for the low energy validity of the effective theory.

The scattering amplitude leading to the result in Eq. \[potential\] includes all diagrams, including box and crossed-box diagrams and some triangle diagrams. In QED or QCD we do not use the full set of diagrams for the running charge, as we include only the vertex corrections and vacuum polarization. In these theories, this is appropriate because it is these diagrams that renormalize the gauge charge. In gravity, none of these diagrams renormalize G at low energy, so the rationale for including only a subset is not clear. Moreover, in gravity, this subset of diagrams does not by itself form a gauge invariant set. Nevertheless, we can look at this subset of diagrams in a particular gauge. In harmonic gauge, the inclusion of both vertex and vacuum polarization would be $$G(q) =G \left[1- \frac{167}{60\pi}G{\mathbf{q}}^2\ln \left( \frac{\mu^2}{{\mathbf q}^2}\right)\right]\,,$$ i.e. it carries the opposite sign from the full result. Even within this subset, one potentially might like to exclude the vertex diagrams because there is no 