Bifurcations in Renormalization Group Flows

Abstract

In this thesis, we show that renormalization group flows generated by the beta functions in a quantum field theory are examples of generic dynamical systems. Using this connection we argue that phase transitions and/or symmetry breaking within renormalization group flows can be represented by bifurcations of the dynamical system. This allows us to apply analytical and numerical methods that have been developed for bifurcation analysis to quantum field theory. We apply these methods to the QCD_4 model with N_c colors and N_f flavors, where we add effective four-fermi interaction to account for non-perturbative terms in the beta functions. We start from the Veneziano limit (N_c,N_f \rightarrow \infty), where this model has been studied before and continue to the low N_c regime. We find that the lower edge of the conformal window is given by a saddle-node bifurcation (or fixed point merger) at constant N_f/N_c for N_c \geq 3. In addition, we find new fixed points in the model. We discuss their relevance and discuss the possibility of phase transitions within the conformal window. Finally, we add a scalar field to the model and show that in the presence of this scalar field there exist similar phase transitions, but the ratio, N_f/N_c, where these transitions occur, changes.