Ward identity that indicates that these must be the same for all particles (i.e. photons vs planets). The vacuum polarization is however universal. Including only it would yield $$G(q) =G \left[1+ \frac{43}{60\pi}G{\mathbf{q}}^2\ln \left( \frac{\mu^2}{{\mathbf q}^2}\right)\right]\,,
\label{purevacpol}$$ as can be seen from Sec. 4. Overall, the non-relativistic scattering amplitude is made up of many large contributions that differ in sign and magnitude. Identifying a subset as the running charge would not capture the leading quantum effects. Moreover, we see that even the sign of the potential definition is not obvious as vertex and vacuum polarization diagrams have opposite signs. The sign of vacuum polarization correction in Eq. \[purevacpol\] agrees with that of the the total scattering amplitude in Eq. \[potential\], but the magnitude is different by a factor of three.

Lessons
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We have explored one-loop calculations in general relativity in the region where there is perturbative control over the theory. There emerges no definition of a running $G$ that is both useful and universal. The nature of the energy expansion of the effective field theory of gravity implies that quantum corrections are associated with the renormalization of higher order operators rather than the original Einstein action. This implies that the 