  In the context of effective field theory, we consider quantum gravity with
minimally coupled massless particles. Fixing the background geometry to be of
the Kerr-Schild type, we fully determine the one-loop effective action of the
theory whose finite non-local part is induced by the long-distance portion of
quantum loops. This is accomplished using the non-local expansion of the heat
kernel in addition to a non-linear completion technique through which the
effective action is expanded in gravitational curvatures. Via Euclidean
methods, we identify a logarithmic correction to the Bekenstein-Hawking entropy
of Schwarzschild black hole. Using dimensional transmutation the result is
shown to exhibit an interesting interplay between the UV and IR properties of
quantum gravity.
