explicit calculation of the sum of direct and radiative cross sections [@Donoghue:1999qh] yields the result $$\begin{aligned}
\label{sum-crs}
&&\hspace{-3em}\left(\frac{d\sigma}{d\Omega}\right)_{tree}
 + \left(\frac{d\sigma}{d\Omega}\right)_{rad.}
  +\left(\frac{d\sigma}{d\Omega}\right)_{nonrad.}=\\
&=& \frac{\kappa^4 s^5}{2048\pi^2 t^2 u^2}\,
\left\{ \rule{0pt}{2.6em}\right.
1 + {\kappa^2 s \over 16 \pi^2}
\left[\rule{0pt}{2em}\right.
 \ln\frac{-t}{s}\ln\frac{-u}{s}+
\frac{tu}{2s^2}\,f\left(\frac{-t}{s},\frac{-u}{s}\right)
\nonumber\\
&&-\left(\frac{t}{s}\,\ln{\frac{-t}{s}}+\frac{u}{s}\,
\ln{\frac{-u}{s}}\right)
\left(
3\ln(2\pi^2)+\gamma+\ln\frac{s}{\mu^2}+
  \frac{\sum_{ij}\eta_i\eta_j{\cal
F}^{(1)}(\gamma_{ij})}{\sum_{ij}\eta_i\eta_j{\cal
F}^{(0)}(\gamma_{ij})}
\right)
\left.\rule{0pt}{2em}\right]
 \left.\rule{0pt}{2.6em}\right\}.\nonumber\end{aligned}$$ Here $\mu$ is an infrared scale related to the experimental energy resolution and ${\cal F}^{(i)}$ are functions defined in [@Donoghue:1999qh] related to the angles of emission of soft graviton radiation.

The last line of the cross section formula is related to infrared physics and does not appear appropriate for inclusion in the definition of a running coupling. Instead we focus on the correction displayed in the preceding line. We would like a renormalization point in the physical region[^3] with a single energy scale $E$. We choose the central physical point $s=2E^2,~ t=u=-E^2$. This leads to the identification $$G^2(E) = G^2\left[1 +\frac{\kappa^2 E^2\left(\ln^2 2+\frac{1}{8}\left(\frac{2297}{180}+\frac{63\pi^2}{64}   \right)  \right)}{8\pi^2}\right]~~~.
\label{running1}$$ We see that this definition leads to a growing running coupling $G(E)$, as opposed to the expectation from asymptotic safety of a decrease in strength at high energy. It works acceptably for this process because it absorbs the main effects of the quantum corrections in 