Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation

TL;DR

This paper reviews and extends the asymptotic safety approach to quantum gravity using Wilsonian renormalization group equations, exploring fixed points, scheme dependencies, and higher derivative gravity effects.

Contribution

It provides a comprehensive analysis of fixed points in quantum gravity with various truncations and schemes, including higher derivative actions and scalar curvature polynomials.

Findings

Identification of a fixed point with three UV-attractive directions in polynomial scalar curvature actions.

Recovery of known asymptotic freedom results for four-derivative gravity terms.

Demonstration of scheme dependence and universality in gravitational couplings.

Abstract

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation. We begin by discussing various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so-called Einstein-Hilbert truncation, where only the cosmological constant and Newton's constant are retained. In this context we make connection with old results, in particular we reproduce the results of the epsilon expansion and the perturbative one loop divergences. We then apply the Renormalization Group to higher derivative gravity. In the case of a general action quadratic in curvature we recover, within certain approximations, the known asymptotic freedom of the four-derivative terms, while Newton's constant and the cosmological constant have a nontrivial fixed point. In the case of actions that are polynomials in the scalar curvature of degree up to eight we find that the theory has a fixed point with three UV-attractive directions, so that the requirement of having a continuum limit constrains the couplings to lie in a three-dimensional subspace, whose equation is explicitly given. We emphasize throughout the difference between scheme-dependent and scheme-independent results, and provide several examples of the fact that only dimensionless couplings can have "universal" behavior.