A proper fixed functional for four-dimensional Quantum Einstein Gravity

TL;DR

This paper constructs and analyzes a fixed functional for four-dimensional Quantum Einstein Gravity using the functional renormalization group, revealing a unique, globally well-defined fixed point that aligns with previous polynomial expansion results.

Contribution

It introduces a novel numerical method to explicitly construct the fixed functional in $f(R)$-gravity, advancing understanding of the non-Gaussian fixed point in Quantum Einstein Gravity.

Findings

The fixed functional is unique and globally well-defined.

In the UV, the solution agrees with polynomial expansion results.

In the IR, the solution scales as $R^2$ with non-analytic terms.

Abstract

Realizing a quantum theory for gravity based on Asymptotic Safety hinges on the existence of a non-Gaussian fixed point of the theory's renormalization group flow. In this work, we use the functional renormalization group equation for the effective average action to study the fixed point underlying Quantum Einstein Gravity at the functional level including an infinite number of scale-dependent coupling constants. We formulate a list of guiding principles underlying the construction of a partial differential equation encoding the scale-dependence of $f(R)$-gravity. We show that this equation admits a unique, globally well-defined fixed functional describing the non-Gaussian fixed point at the level of functions of the scalar curvature. This solution is constructed explicitly via a numerical double-shooting method. In the UV, this solution is in good agreement with results from polynomial expansions including a finite number of coupling constants, while it scales proportional to $R^2$, dressed up with non-analytic terms, in the IR. We demonstrate that its structure is mainly governed by the conformal sector of the flow equation. The relation of our work to previous, partial constructions of similar scaling solutions is discussed.