for $R^2$ and $R_{\alpha\beta}R^{\alpha\beta}$ respectively.

The polarization tensor can be written in the form [@Weinberg:1980gg] $$\begin{aligned}
\Pi^{\alpha\beta,\mu\nu}(q)&=&q^4A(q^2)L^{\alpha\beta}(q)L^{\mu\nu}(q)-q^2B(q^2)\left[L^{\alpha\mu}(q)L^{\beta\nu}(q)+L^{\alpha\nu}(q)L^{\beta\mu}(q) -2L^{\alpha\beta}(q)L^{\mu\nu}(q)\right],\end{aligned}$$ where $L^{\mu\nu}(q)=\eta^{\mu\nu}-q^{\mu}q^{\nu}/q^2$. Contracting $\Pi^{\alpha\beta,\gamma\delta}(q)$ with $L^{\alpha\beta}(q)L^{\mu\nu}(q)$, and $L^{\alpha\mu}(q)L^{\beta\nu}(q)$ we obtain two equations in $A(q^2)$, and $B(q^2)$ $$\begin{aligned}
\nonumber
\Pi^{\alpha\beta,\mu\nu}(q)L_{\alpha\beta}(q)L_{\mu\nu}(q)&=&3q^2\left[3q^2A(q^2)+4B(q^2) \right]\,,\\
\Pi^{\alpha\beta,\mu\nu}(q)L_{\alpha\mu}(q)L_{\beta\nu}(q)&=&3q^2\left[q^2A(q^2)-2B(q^2) \right]\,.\end{aligned}$$ Hence, we find $$\begin{aligned}
\nonumber
A(q^2)&=&-\frac{1}{30\pi}G\ln\left(\frac{-q^2}{\mu_1^2}\right)-\frac{7}{10\pi}G\ln\left(\frac{-q^2}{\mu_2^2}\right)\,,\\
B(q^2)&=&\frac{7}{40\pi}Gq^2\ln\left(\frac{-q^2}{\mu_2^2}\right)\,.\end{aligned}$$

On the other hand, the bare propagator takes the general form $$\begin{aligned}
i{\cal D}^{\alpha\beta,\mu\nu}(q^2)=\frac{i}{2q^2}\left[L^{\alpha\mu}L^{\beta\nu}+L^{\alpha\nu}L^{\beta\mu}-L^{\alpha\beta}L^{\mu\nu} \right]\,,\end{aligned}$$ while the quantum corrected propagator reads $$\begin{aligned}
\nonumber
i{\cal D}'^{\alpha\beta,\mu\nu}&=&i{\cal D}^{\alpha\beta,\mu\nu}+i{\cal D}^{\alpha\beta,\gamma\delta}i\Pi_{\gamma\delta,\rho\tau}i{\cal D}^{\rho\tau,\mu\nu}\\
&=&\frac{i}{2q^2}(1+2B(q^2)) \left[L^{\alpha\mu}L^{\beta\nu}+L^{\alpha\nu}L^{\beta\mu}-L^{\alpha\beta}L^{\mu\nu} \right]-i\frac{A(q^2)}{4}L^{\alpha\beta}L^{\mu\nu}\,.
\label{dressed}\end{aligned}$$ The first term above is a dressed propagator. Therefore, it is appropriate to define the running coupling as $$\begin{aligned}
\label{one way of normalizing}
 G(q^2)=G(1+2B(q^2))=G\left(1+\frac{7}{20\pi}Gq^2\ln\left(\frac{-q^2}{\mu_2^2}\right)\right)\,.\end{aligned}$$ On the other hand, the second term in Eq. \[dressed\] also contributes comparably to gravitational amplitudes. For example, non-relativistic scattering involves the $00,00$ component of ${\cal D}^{\mu\nu,\alpha\beta}$ and we can equally define a running coupling from that component $$\begin{aligned}
{\cal D}^{00,00}=\frac{1}{2q^2}\left(1+2B-\frac{q^2A}{2}  \right)L^{00}L^{00}=\frac{1}{2q^2} G(q^2)L^{00}L^{00}\,,\end{aligned}$$ and hence, $$\begin{aligned}
\label{0000 components}
G(q^2)=G\left[1+\frac{1}{60\pi} Gq^2\ln\left(\frac{-q^2}{\mu_2^2}\right)+\frac{7}{10\pi} Gq^2\ln\left(\frac{-q^2}{\mu_1^2}\right) \right]\,.\end{aligned}$$ This corresponds exactly to the vacuum polarization contribution to the shift in the Newtonian interaction as calculated in [@Donoghue:1994dn], but it could be more widely applicable. Since $M_P$ is the only scale in gravity, we expect $\mu_1$ and $\mu_2$ to be of this order, and 