of the reaction $A+B\to A+B$ are $$\begin{aligned}
\nonumber
{\cal M}_{tree}&=&\frac{i\kappa^2 su}{4t}\,,\\
\nonumber
{\cal M}_{1-loop}&=&i\frac{\kappa^4}{\left(4\pi \right)^2}\left[\frac{1}{16}\left(s^4I_4(s,t)+u^4 I_4(u,t)\right) +\frac{1}{8}\left(s^3+u^3+tsu \right)I_3(t)-\frac{1}{8}\left(s^3I_3(s)+u^3I_3(u)  \right)  \right.\\
\label{ABAB scattering}
&&\left.\quad\quad\quad-\frac{1}{240}\left(71us-11t^2\right)I_2(t)+\frac{1}{16}\left(s^2I_2(s)+u^2I_2(u)\right)  \right]\,.\end{aligned}$$ Then, we use Eq. \[removing IR\] in removing the IR divergences. The resulting hard amplitude reads $$\begin{aligned}
\nonumber
{\cal M}_{h}&=&i\frac{\kappa^4}{\left(4\pi \right)^2}\left[\frac{1}{8}\left(\frac{s^3}{t}\ln(-s)\ln(-t)+\frac{u^3}{t} \ln(-u)\ln(-t)\right) -\frac{1}{16t}\left(s^3+u^3+tsu \right)\ln(-t)+\frac{1}{16}\left(s^2\ln^2(-s)+u^2\ln^2(-u)\right)     \right.\\
\nonumber
&&\left.\quad\quad\quad+\frac{us}{16t}\left(s\ln^2(-s)+t\ln^2(-t)+u\ln^2(-u)  \right)+\frac{1}{240}\left(71us-11t^2\right)\ln(-t)-\frac{1}{16}\left(s^2\ln(-s)+u^2\ln(-u)\right) \right]\,,\\\end{aligned}$$ and the total amplitude at the center kinematic point $s=2E^2,~t=u=-E^2$ is $$\begin{aligned}
{\cal M}_{total}=\frac{i\kappa^2E^2}{2}\left[1-\frac{\kappa^2 E^2}{10(4\pi)^2}\left(\left(19+10\ln 2\right)\ln\left(\frac{E^2}{\mu^2}\right)+5\left(\pi^2-(\ln 2-1)\ln 2  \right) \right)  \right]\,.\end{aligned}$$

On the other hand, the amplitude of the reaction $A+A\to B+B$ is given by Eq. \[ABAB scattering\] with the exchange $s\leftrightarrow t$, and has the amplitude $$\begin{aligned}
{\cal M}_{total}=\frac{i\kappa^2E^2}{8}\left[1+\frac{\kappa^2 E^2}{10(4\pi)^2}\left(9\ln\left(\frac{E^2}{\mu^2}\right)-5\pi^2+\left(19+5\ln 2  \right)\ln2 \right)  \right]\,.\end{aligned}$$ The crossing problem is obvious here. The loop corrections are in opposite directions in the two reactions, largely because of the change in sign of the kinematic variables under crossing. Any definition of a running $G$ cannot capture this behavior - the coupling must either increase with energy scale or decrease with energy. The processes also illustrate the non-universality problem. Even two reactions that are this closely related have a different magnitude for the one loop correction when evaluated in the physical region.

Gravitational scattering of heavy masses
========================================

Finally we consider 