  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 renormalization group equation. This method relies only
on the local Schwinger-DeWitt expansion of the heat kernel and allows for a
re-summation of leading large logarithms 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} \ll \nabla^{2} R $ 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 $R^{2} \gg \nabla^{2} R$, which is often of interest in
cosmology.
