is graviton-graviton scattering. The lowest order scattering amplitude involves a large number of individual tree diagrams but is given by the simple form $${A}^{tree}(++;++) = i\frac{\kappa^2}{4} \frac{s^3}{tu}\,,
\label{gravtree}$$ where the signs $+,-$ refer to helicity indices and $s,t,u$ are the usual Mandelstam variables. In power counting, this is a dimensionless amplitude of order $GE^2$. Our labeling of momentum and helicities corresponds to the final state particle being outgoing, in contrast to some conventions in the literature which label all particles as ingoing[^2].

The one loop amplitudes have been calculated by Dunbar and Norridge [@Dunbar:1994bn]. These are of order $G^2E^4$ and take the form

$$\begin{aligned}
\label{eq:2}
{\cal A}^{1-loop}(++;--) & = & -i\,{\kappa^4 \over 30720
\pi^2}
\left( s^2+t^2 + u^2 \right) \,,  \nonumber\\
{\cal A}^{1-loop}(++;+-) & = & -{1 \over 3}
{\cal A}^{1-loop}(++;--)\nonumber \\
{\cal A}^{1-loop}(++;++) & = &\frac{\kappa^2}
{4(4\pi)^{2-\epsilon}}\,
 \frac{\Gamma^2(1-\epsilon)\Gamma(1+\epsilon)}
 {\Gamma(1-2\epsilon)}\,
 {\cal A}^{tree}(++;++)\,\times(s\,t\,u)\\
&&\hspace{-0em}\times\left[\rule{0pt}{4.5ex}\right.
\frac{2}{\epsilon}\left(
\frac{\ln(-u)}{st}\,+\,\frac{\ln(-t)}{su}\,+\,\frac{\ln(-s)}{tu}
\right)+\,\frac{1}{s^2}\,f\left(\frac{-t}{s},\frac{-u}{s}\right)
\nonumber\\&&\hspace{1.4em}
+2\,\left(\frac{\ln(-u)\ln(-s)}{su}\,+\,\frac{\ln(-t)\ln(-s)}{tu}\,+\,
\frac{\ln(-t)\ln(-s)}{ts}\right)
\left.\rule{0pt}{4.5ex}\right]\,,\nonumber\end{aligned}$$

where $$\begin{aligned}
\label{eq:f}
f\left(\frac{-t}{s},\frac{-u}{s}\right)&=&
\frac{(t+2u)(2t+u)\left(2t^4+2t^3u-t^2u^2+2tu^3+2u^4\right)}
{s^6}
\left(\ln^2\frac{t}{u}+\pi^2\right)\nonumber\\&&
+\frac{(t-u)\left(341t^4+1609t^3u+2566t^2u^2+1609tu^3+
341u^4\right)}
{30s^5}\ln\frac{t}{u}\nonumber\\&&
+\frac{1922t^4+9143t^3u+14622t^2u^2+9143tu^3+1922u^4}
{180s^4}\,.\end{aligned}$$ Other amplitudes can be obtained from these by crossing.

The dimensional regularization parameter $\epsilon =(4-d)/2$ appears in the amplitude ${\cal A}^{1-loop}(++;++)$. This is an infrared divergence, and is canceled as usual from the radiation of soft gravitons. An 