Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity

TL;DR

This work develops and applies the effective average action, a continuum Wilsonian RG tool, to Quantum Einstein Gravity (QEG). By formulating the FRGE within a background-field framework and employing truncations such as the Einstein–Hilbert and $R^2$ actions, it provides nonperturbative evidence for a non-Gaussian fixed point (NGFP) with a finite UV critical hypersurface, supporting the asymptotic safety of gravity. The results show the NGFP is robust across regulator schemes and gauge choices, with a universal product $g^*\lambda^*$ and UV-attractive directions, suggesting QEG could be nonperturbatively renormalizable. The framework also links to phenomenology through scale-dependent spacetimes and RG-improvement, and aligns with independent truncation studies that reinforce the asymptotic-safety scenario for four-dimensional gravity.

Abstract

These lecture notes provide a pedagogical introduction to a specific continuum implementation of the Wilsonian renormalization group, the effective average action. Its general properties and, in particular, its functional renormalization group equation are explained in a simple scalar setting. The approach is then applied to Quantum Einstein Gravity (QEG). The possibility of constructing a fundamental theory of quantum gravity in the framework of Asymptotic Safety is discussed and the supporting evidence is summarized.

Figures (7)

• Figure 1: The points of theory space are the action functionals $A[\, \cdot \,]$. The RG equation defines a vector field $\vec{\beta}$ on this space; its integral curves are the RG trajectories $k \mapsto \Gamma_k$. They start at the bare action $S$ and end at the standard effective action $\Gamma$.
• Figure 2: Schematic picture of the UV critical hypersurface ${\cal S}_{\rm UV}$ of the NGFP. It is spanned by RG trajectories emanating from the NGFP as the RG scale $k$ is lowered. Trajectories not in the surface are attracted towards ${\cal S}_{\rm UV}$ as $k$ decreases. (The arrows point in the direction of decreasing $k$, from the "UV" to the "IR".)
• Figure 3: RG flow in the $g$-$\lambda-$plane. The arrows point in the direction of increasing coarse graining, i.e., of decreasing $k$. (From frank1.)
• Figure 4: Comparison of $\lambda^*,g^*, \theta^{\prime}$ and $\theta^{\prime \prime }$ for different cutoff functions in dependence of the dimension $d$. Two versions of the sharp cutoff (sc) and the exponential cutoff with $s=1$ (Exp) have been employed. The upper line shows that for $2+ \epsilon \le d \le 4$ the cutoff scheme dependence of the results is rather small. The lower diagram shows that increasing $d$ beyond about 5 leads to a significant difference in the results for $\theta^{\prime}, \theta^{\prime \prime}$ obtained with the different cutoff schemes. (From frank1.)
• Figure 5: (a) $g^*$, $\lambda^*$, and $g^*\lambda^*$ as functions of $s$ for $1\le s\le 5$, and (b) $\beta^*$ as a function of $s$ for $1\le s\le 30$, using the family of exponential shape functions (\ref{['G19']}). (From ref. oliver3.)