Path integral measure and RG equations for gravity

Considering the Einstein-Hilbert truncation for the running action in (Euclidean) quantum gravity, we derive the renormalization group equations for the cosmological and Newton constant. We find that these equations admit only the Gaussian fixed point with a UV-attractive and a UV-repulsive eigendirection, and that there is no sign of the nontrivial UV-attractive fixed point of the asymptotic safety scenario. Crucial to our analysis is a careful treatment of the measure in the path integral that defines the running action and a proper introduction of the physical running scale <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>k</a:mi></a:math>. We also show why and how in usual implementations of the RG equations the aforementioned UV-attractive fixed point is generated.