# Composite higher derivative operators in $d=2+\epsilon$ dimensions and the spectrum of asymptotically safe gravity

## TL;DR

This paper investigates the renormalization and spectrum of higher derivative operators in quantum gravity near two dimensions, confirming the asymptotic safety scenario and identifying relevant operators at the fixed point.

## Contribution

It introduces a scheme- and gauge-independent approach to analyze composite operators in quantum gravity and determines their scaling spectrum at the fixed point.

## Key findings

- Existence of an ultraviolet fixed point for Newton's constant.
- Identification of a relevant operator near four dimensions.
- Other previously known operators are marginal or trivial on-shell.

## Abstract

We discuss the renormalization of Einstein-Hilbert's gravity in $d=2+\epsilon$ dimensions. We show that the application of the path-integral approach leads naturally to scheme- and gauge-independent results on-shell, but also gives a natural notion of quantum metric off-shell, which is the natural argument of the effective action, even at the leading order in perturbation theory. The renormalization group of Newton's constant is consistent with the asymptotic safety scenario for quantum gravity in that it has an ultraviolet relevant fixed point. We extend the approach to the analysis of curvature square operators, understood as composites operators, which allows for the determination of the spectrum of scaling operators at the scale invariant fixed point. The analysis suggests that there is one operator that becomes relevant close to $d=4$ dimensions, while other operators previously found in the literature are either marginal or trivial on-shell.