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

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 ${\lambda }_n\propto bn\ln n$. 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.

arXiv comment35 pages, no figures; minor amendments. version published in JHEP

References in corpus (48)

• Exact evolution equation for the effective potential
• Nonperturbative Evolution Equation for Quantum Gravity
• The Exact Renormalisation Group and Approximate Solutions
• Optimised Renormalisation Group Flows
• Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation
• Flow Equation of Quantum Einstein Gravity in a Higher-Derivative Truncation
• Derivative Expansion of the Exact Renormalization Group
• Ultraviolet properties of f(R)-Gravity
• On the renormalization group flow of f(R)-gravity
• Elements of the Continuous Renormalization Group
• On Truncations of the Exact Renormalization Group
• Asymptotic safety in the f(R) approximation
• Search of scaling solutions in scalar-tensor gravity
• The Renormalization Group flow of unimodular f(R) gravity
• Asymptotic safety of quantum gravity beyond Ricci scalars
• A flow equation for f(R) gravity and some of its exact solutions
• Aspects of asymptotic safety for quantum gravity
• Renormalization Group Equation and scaling solutions for f(R) gravity in exponential parametrization
• A proper fixed functional for four-dimensional Quantum Einstein Gravity
• Asymptotic safety in $O(N)$ scalar models coupled to gravity
• Derivative expansion of the renormalization group in O(N) scalar field theory
• Black hole thermodynamics under the microscope
• Three dimensional massive scalar field theory and the derivative expansion of the renormalization group
• On the number of relevant operators in asymptotically safe gravity
• The Renormalization Group and Two Dimensional Multicritical Effective Scalar Field Theory
• Renormalization Group Equation for $f(R)$ gravity on hyperbolic spaces
• Fixed-Point Structure of Scalar Fields
• Fixed-Functionals of three-dimensional Quantum Einstein Gravity
• RG flows of Quantum Einstein Gravity on maximally symmetric spaces
• Large curvature and background scale independence in single-metric approximations to asymptotic safety
• Background independent exact renormalization group for conformally reduced gravity
• The background scale Ward identity in quantum gravity
• On the Fixed-Point Structure of Scalar Fields
• Asymptotic solutions in asymptotic safety
• Fixed point structure of the conformal factor field in quantum gravity
• Asymptotically Safe $f(R)$-Gravity Coupled to Matter II: Global Solutions
• Background Scale Independence in Quantum Gravity
• Non-Compact Pure Gauge QED in 3D is Free
• Fixed Points of Quantum Gravity and the Dimensionality of the UV Critical Surface
• Curvature dependence of quantum gravity with scalars
• The fate of non-polynomial interactions in scalar field theory
• Quantum gravity, renormalizability and diffeomorphism invariance
• Renormalization group properties of the conformal sector: towards perturbatively renormalizable quantum gravity
• Perturbatively renormalizable quantum gravity
• Renormalization group properties of the conformal mode of a torus
• The continuum limit of quantum gravity at first order in perturbation theory
• Fixed Functionals in Asymptotically Safe Gravity
• The continuum limit of quantum gravity at second order in perturbation theory

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