Renormalization of multicritical scalar models in curved space

We consider the leading order perturbative renormalization of the multicritical $\phi^{2n}$ models and some generalizations in curved space. We pay particular attention to the nonminimal interaction with the scalar curvature $\frac{1}{2}\xi \phi^2 R$ and discuss the emergence of the conformal value of the coupling $\xi$ as the renormalization group fixed point of its beta function at and below the upper critical dimension as a function of $n$. We also examine our results in relation with Kawai and Ninomiya's formulation of two dimensional gravity.Comment: 13 pages, 3 figures; v3: matches the published versio

Selected applications of functional RG

In this thesis we will address the study of quantum field theories using the exact renormalization group technique. In particular, we will calculate the flow of a Yukawa system coupled to gravity and that of a higher derivative nonlinear sigma model. The study of the Yukawa system in presence of gravity, as well as the study of any matter theory coupled to gravity, is important for two reason. First, it is interesting to see what gravitational dressing one should expect to the beta functions of any matter theory. Second, it is important to test the possibility that gravity is an asymptotically safe theory [1, 2] against the addition of matter degrees of freedom. We also calculate the 1-loop flow of a general higher derivative nonlinear sigma model, using exact renormalization group techniques. We think that the nonlinear sigma model is an important arena to test the exact renormalization. The reason is that the nonlinear sigma model shares many of the features of gravity, like perturbative nonrenormalizability, but does not have the additional complication of a local gauge invariance. Furthermore, it is an interesting question whether a nonlinear sigma model admits a ultraviolet limit or it has to be regarded as an effective field theory only. The plan of the work is as follows. In Chapter 1 we give a very brief introduction to the technique of functional exact renormalization group. In Chapter 2 we introduce the notion of \u201cAsymptotic Safety\u201d [1] and discuss some of the approximation schemes generally involved in calculations. In Chapter 3 we use a simple Yukawa model as a toy model for many of the techniques we will need later. We also discuss the background field method in the context of a theory with local gauge invariance, which will turn out to be useful in Chapter 4. In Chapter 4 we couple the simple Yukawa model with gravity and calculate its renormalization group flow. In Chapter 5 we study numerically the flow calculated in Chapter 4 and point out the possibility that the model admits a nontrivial ultraviolet limit. Chapter 6 is the final chapter and contains the study of the flow of the higher derivative nonlinear sigma model; it is a self contained chapter. In fact, Chapter 5 and 6 contain separate discussions for the results of the Yukawa and sigma model, respectively. We dedicate the appendices to arguments that would have implied very long digressions in the main text

On the non-local heat kernel expansion

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 curvature. Our method can be generalized easily to the derivation of the non-local heat kernel expansion of a wide class of differential operators.Comment: 23 pages, 1 figure, 31 diagrams; references added; to appear in JM

RG flows of Quantum Einstein Gravity on maximally symmetric spaces

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.Comment: 35 pages, 6 figure

Fixed-Functionals of three-dimensional Quantum Einstein Gravity

We study the non-perturbative renormalization group flow of f(R)-gravity in three-dimensional Asymptotically Safe Quantum Einstein Gravity. Within the conformally reduced approximation, we derive an exact partial differential equation governing the RG-scale dependence of the function f(R). This equation is shown to possess two isolated and one continuous one-parameter family of scale-independent, regular solutions which constitute the natural generalization of RG fixed points to the realm of infinite-dimensional theory spaces. All solutions are bounded from below and give rise to positive definite kinetic terms. Moreover, they admit either one or two UV-relevant deformations, indicating that the corresponding UV-critical hypersurfaces remain finite dimensional despite the inclusion of an infinite number of coupling constants. The impact of our findings on the gravitational Asymptotic Safety program and its connection to new massive gravity is briefly discussed.Comment: 34 pages, 14 figure

Scheme dependence and universality in the functional renormalization group

We prove that the functional renormalization group flow equation admits a perturbative solution and show explicitly the scheme transformation that relates it to the standard schemes of perturbation theory. We then define a universal scheme within the functional renormalization group.Comment: 5 pages, improved version; v2: published version; v3 and v4: fixed various typos (final result is unaffected

Scaling and superscaling solutions from the functional renormalization group

We study the renormalization group flow of $\mathbb{Z}_2$-invariant supersymmetric and non-supersymmetric scalar models in the local potential approximation using functional renormalization group methods. We focus our attention to the fixed points of the renormalization group flow of these models, which emerge as scaling solutions. In two dimensions these solutions are interpreted as the minimal (supersymmetric) models of conformal field theory, while in three dimension they are manifestations of the Wilson-Fisher universality class and its supersymmetric counterpart. We also study the analytically continued flow in fractal dimensions between 2 and 4 and determine the critical dimensions for which irrelevant operators become relevant and change the universality class of the scaling solution. We also include novel analytic and numerical investigations of the properties that determine the occurrence of the scaling solutions within the method. For each solution we offer new techniques to compute the spectrum of the deformations and obtain the corresponding critical exponents.Comment: 23 pages, 14 figures; v2: several improvements, new references, version to appear in PR

A functional perspective on emergent supersymmetry

We investigate the emergence of ${\cal N}=1$ supersymmetry in the long-range behavior of three-dimensional parity-symmetric Yukawa systems. We discuss a renormalization approach that manifestly preserves supersymmetry whenever such symmetry is realized, and use it to prove that supersymmetry-breaking operators are irrelevant, thus proving that such operators are suppressed in the infrared. All our findings are illustrated with the aid of the $\epsilon$-expansion and a functional variant of perturbation theory, but we provide numerical estimates of critical exponents that are based on the non-perturbative functional renormalization group.Comment: 28 pages, 2 figures; v2: published version, includes a new appendi

Leading order CFT analysis of multi-scalar theories in d>2

We investigate multi-field multicritical scalar theories using CFT constraints on two- and three-point functions combined with the Schwinger-Dyson equation. This is done in general and without assuming any symmetry for the models, which we just define to admit a Landau-Ginzburg description that includes the most general critical interactions built from monomials of the form $\phi_{i_1} \cdots \phi_{i_m}$. For all such models we analyze to the leading order of the $\epsilon$-expansion the anomalous dimensions of the fields and those of the composite quadratic operators. For models with even $m$ we extend the analysis to an infinite tower of composite operators of arbitrary order. The results are supplemented by the computation of some families of structure constants. We also find the equations which constrain the nontrivial critical theories at leading order and show that they coincide with the ones obtained with functional perturbative RG methods. This is done for the case $m=3$ as well as for all the even models. We ultimately specialize to $S_q$ symmetric models, which are related to the $q$-state Potts universality class, and focus on three realizations appearing below the upper critical dimensions $6$, $4$ and $\frac{10}{3}$, which can thus be nontrivial CFTs in three dimensions.Comment: 58 pages; v2: minor clarifications added, to appear in EPJ