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 “Asymptotic Safety” [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.