Nonlocal quantum effective actions in Weyl-Flat spacetimes

Virtual massless particles in quantum loops lead to nonlocal effects which can have interesting consequences, for example, for primordial magnetogenesis in cosmology or for computing finite N corrections in holography. We describe how the quantum effective actions summarizing these effects can be computed efficiently for Weyl-flat metrics by integrating the Weyl anomaly or, equivalently, the local renormnalization group equation. This method relies only on the local Schwinger-DeWitt expansion of the heat kernel and allows for a re-summation of the anomalous leading large logarithms of the scale factor, log a(x), in situations where the Weyl factor changes by several e-foldings. As an illustration, we obtain the quantum effective action for the Yang-Mills field coupled to massless matter, and the self-interacting massless scalar field. Our action reduces to the nonlocal action obtained using the Barvinsky-Vilkovisky covariant perturbation theory in the regime R-2 << del R-2 for a typical curvature scale R, but has a greater range of validity effectively re-summing the covariant perturbation theory to all orders in curvatures. In particular, it is applicable also in the opposite regime R-2 << del R-2, which is often of interest in cosmology.