\phi +\phi$ via graviton exchange at tree level has $s,~t, ~{\rm and}~ u$ channel poles, with amplitude $$\begin{aligned}
{\cal M}_{tree}=i\frac{\kappa^2}{4}\left[\frac{st}{u}+\frac{su}{t}+\frac{tu}{s} \right]\,.\end{aligned}$$ We note that this amplitude, and the loop amplitudes to follow, has a permutation (crossing) symmetry such that all channels are governed by the same amplitude. This will eliminate the crossing problem that arises in most other reactions. However we can test for universality by testing whether the interactions lead to a similar running coupling as suggested in the purely gravitational sector.

In this theory there is a higher order operator which is required at one loop. Divergences proportional to $${\cal L}_2 = \frac{203}{320\epsilon} (D_\mu \phi D^\mu \phi)^2$$ arise at one loop. In matrix elements, this operator generates a contribution proportional to $s^2 + t^2 +u^2$. The one loop amplitudes, up to rational terms in the kinematic variables, have been given in [@Dunbar:1995ed]. However the rational terms are constrained by the permutation symmetry to also be proportional to $s^2 + t^2 +u^2$ and we will absorb them into the higher order Lagrangian ${\cal L}_2$.

The total scattering amplitude of this process, apart from a polynomial in $s$, $t$, and $u$, is given by [^4] $$\begin{aligned}
\nonumber
{\cal M}_{1-loop}&=&i\frac{\kappa^4}{\left(4\pi\right)^2}\left[\frac{(s^4+t^4)}{16}I_4(s,t)+\frac{(s^4+u^4)}{16}I_4(s,u)+\frac{(u^4+t^4)}{16}I_4(t,u)-\frac{s(s^2+2t^2+2u^2)}{8}I_3(s)  \right.\\
