Provable properties of asymptotic safety in $f(R)$ approximation

TL;DR

This paper investigates the properties of fixed points and eigenoperators in an $f(R)$ approximation to asymptotic safety, revealing universal and cutoff-dependent features, and identifying a discrete spectrum of relevant operators.

Contribution

It provides a rigorous analysis of fixed points and eigenoperator spectra in $f(R)$ asymptotic safety, highlighting universality and cutoff dependence, and introduces novel operators in the conformal sector.

Findings

Large $n$ eigenvalues scale as $b n \, \ln n$ with non-universal $b$

Fixed points are discrete for right-sign cutoff, forming a quantized spectrum

Wrong-sign cutoff yields a continuum of fixed points unless square-integrability is imposed

Abstract

We study an $f(R)$ approximation to asymptotic safety, using a family of non-adaptive cutoffs, kept general to test for universality. Matching solutions on the four-dimensional sphere and hyperboloid, we prove properties of any such global fixed point solution and its eigenoperators. For this family of cutoffs, the scaling dimension at large $n$ of the $n^\text{th}$ eigenoperator, is $\lambda_n\propto b\, n\ln n$. The coefficient $b$ is non-universal, a consequence of the single-metric approximation. The large $R$ limit is universal on the hyperboloid, but not on the sphere where cutoff dependence results from certain zero modes. For right-sign conformal mode cutoff, the fixed points form at most a discrete set. The eigenoperator spectrum is quantised. They are square integrable under the Sturm-Liouville weight. For wrong sign cutoff, the fixed points form a continuum, and so do the eigenoperators unless we impose square-integrability. If we do this, we get a discrete tower of operators, infinitely many of which are relevant. These are $f(R)$ analogues of novel operators in the conformal sector which were used recently to furnish an alternative quantisation of gravity.