which appears in Eq. \[running2\] could reasonably be associated with the higher order operator $d_1$, and perhaps should be removed from this expression. The most serious flaw of this result in comparison to Eq. \[running1\] is that it has the opposite sign. The leading corrections to graviton scattering and to scalar scattering go in the opposite direction, which is not accountable by a common definition of a running coupling, an obvious lack of universality.

Gravitational scattering of non-identical particles
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Here we consider a different situation for the matter couplings - the scattering of non-identical particles. We will neglect the particle masses, so this corresponds to scattering at $s>>m^2$. This situation demonstrates both the crossing problem and the non-universality problem.

The example of the last section has more crossing symmetry than most gravitational reactions. Processes involving non-identical particles, or with fermions, will typically involve dominantly only one of the $s,t,u$ channels. Typical gravitational scattering of very massive particles will involve primarily $t$-channel exchange. Such distinctions highlight the difficulty of any given definition of a running $G$ being applicable to all processes.

By a direct computation of the appropriate set of Feynman diagrams, we find that the tree and one-loop amplitudes 