We compute the gravitational corrections to the running of couplings in a scalar-fermion system, using the Wilsonian approach. Our discussion is relevant for symmetric as well as for broken scalar phases. We find that the Yukawa and quartic scalar couplings become irrelevant at the Gaussian fixed point.

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 R², 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.

We use the Wetterich-equation to study the renormalization group flow of f (R)-gravity in a three-dimensional, conformally reduced setting. Building on the exact heat kernel for maximally symmetric spaces, we obtain a partial differential equation which captures the scale-dependence of f (R) for positive and, for the first time, negative scalar curvature. The effects of different background topologies are studied in detail and it is shown that they affect the gravitational RG flow in a way that is not visible in finite-dimensional truncations. Thus, while featuring local background independence, the functional renormalization group equation is sensitive to the topological properties of the background. The detailed analytical and numerical analysis of the partial differential equation reveals two globally well-defined fixed functionals with at most a finite number of relevant deformations. Their properties are remarkably similar to two of the fixed points identified within the R ²-truncation of full Quantum Einstein Gravity. As a byproduct, we obtain a nice illustration of how the functional renormalization group realizes the “integrating out” of fluctuation modes on the three-sphere.

We propose a novel derivation of the non-local heat kernel expansion, first studied by Barvinsky, Vilkovisky, and Avramidi, based on simple diagrammatic equations satisfied by the heat kernel. For Laplace-type differential operators, we obtain the explicit form of the non-local heat kernel form factors to second order in the curvatures. Our method can be generalized easily to the derivation of the non-local heat kernel expansion of a wide class of differential operators.

Within the functional renormalization group approach we study the effective quantum field theory of Einstein gravity and one self-interacting scalar coupled to Nf Dirac fermions. We include in our analysis the matter anomalous dimensions induced by all the interactions and analyze the highly nonlinear beta functions determining the renormalization flow. We find the existence of a nontrivial fixed point structure both for the gravity and the matter sector, besides the usual Gaussian matter one. This suggests that asymptotic safety could be realized in the gravitational sector and in the standard model. Nontriviality in the Higgs sector might involve gravitational interactions.