are not present. For logarithmic running couplings, the kinematic ambiguity is absent because the real part of $\ln q^2$ is the same for both spacelike and timelike $q^2$. The process dependence is absent because the running is connected with the actual renormalization of the charge - the $\ln (q^2/\mu^2)$ factor that arises is tied to the $1/\epsilon$ divergence absorbed into the renormalized charge. Because charge renormalization is universal, the corresponding logarithmic corrections are also universal.

In [@Anber:2010uj], we did find one case where a gravitational correction to a running coupling could be constructed perturbatively without obvious flaws. This was $\lambda \phi^4$ theory. In this case, the unique higher order operator vanishes by the equation of motion - the gravitational corrections are one-loop-finite. In addition, the complete permutation symmetry of the $\phi^4$ interaction means that all reactions are crossing symmetric, and involve a unique crossing symmetric combination of kinematic invariants. There is no crossing problem. Because there is only one type of vertex in this theory, there is also no non-universality problem. These allowed a reasonable definition of the gravitational correction to the running of $\lambda$. However, we will see that these nice properties are not shared by the gravitational self 