Our two Mohameds and I have recently posted a paper on the gravitational effect on running couplings. There has been a lot of interest in this effect since a paper by Robinson and Wilczek. Our paper is a set of explicit calculations to see how useful the idea of a running coupling is in a well defined setting involving scattering processes. However our conclusion is that the idea of a running coupling in this setting is not useful, because even the sign of the correction can change as one goes from space-like processes to time-like ones.
Instead, the dominant physics is that of operator mixing rather than the evolution of a single coupling constant.
Our work was performed in the perturbative regime, so we had control over the techniques, and we explored a variety of possibilities for the definition of a running coupling. We note that there has been a lot of work lately on the related topic of the “asymptotic safety” program, which takes a running coupling into the non-perturbative region beyond the Planck scale. We were careful not to say anything about the asymptotic safety program, but a reader will likely recognize that our results are not positive for that research topic. We hope to address this more explicitly in the future.