Path integral measure and RG equations for gravity

TL;DR

This paper derives the RG equations for quantum gravity using the Einstein-Hilbert truncation, revealing only the Gaussian fixed point and challenging the existence of a non-trivial UV fixed point in the asymptotic safety scenario.

Contribution

It provides a careful analysis of the path integral measure and the physical scale, showing the absence of a non-trivial UV fixed point in this truncation.

Findings

Only Gaussian fixed point found with UV-attractive and UV-repulsive directions.

No evidence of non-trivial UV fixed point in Einstein-Hilbert truncation.

Clarifies the role of the measure and scale in RG equations.

Abstract

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 non-trivial 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 $k$. We also show why and how in usual implementations of the RG equations the aforementioned UV-attractive fixed point is generated.