Lectures on the functional renormalization group method

TL;DR

This work presents a comprehensive introduction to the functional renormalization group (FRG) as a general, nonperturbative framework for solving strongly coupled quantum field theories beyond critical points. It surveys two core FRG formalisms—the Wegner–Houghton (WH) blocking in continuous space-time and Polchinski’s equation with a smooth cutoff—along with their gradient expansions, local potential approximation, and composite-operator renormalization. The text develops both loop-resummation and tree-level (saddle-point) contributions, extends the analysis to continuous evolution and internal-space blocking, and applies the formalism to fixed points, global RG flows, and concrete models such as the sine-Gordon model and gauge theories, highlighting instability-induced renormalization and the condensation-as-crossover phenomenon. A key message is that FRG provides a robust, unifying language to track the scale dependence of many coupling constants and operators, enabling nonperturbative insights and cross-disciplinary applications, including potential implications for the structure of vacua in gauge theories and beyond. The work also emphasizes the geometric interpretation of operator mixing as parallel transport along RG trajectories, and discusses the practical limitations and future directions for achieving a genuinely nonperturbative, gauge-invariant FRG framework across diverse physical systems.

Abstract

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained in a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scalar theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

Figures (11)

• Figure 1: Graphs contributing to the blocking, (a): tree-level, (b): one-loop corrections. The dashed line stands for a particle of momentum $|p|=k$ and the solid lines represent $\phi$.
• Figure 2: The first four one-loop graphs contributing to the WH equation in the local potential approximation when the potential $U(\phi)$ is truncated to the terms $\phi^2$ and $\phi^4$. The dashed line corresponds a particle of momentum $|p|=k$.
• Figure 3: Graphs contributing to $\beta_n$, (a): $n=2$, (b): $n=3$ and (c): $n=4$.
• Figure 4: A two-loop contribution to $g_2$.
• Figure 5: An U.V. fixed point and its vicinity. The $x$ and the $y$ axis correspond a relevant and an irrelevant operator, respectively. The circle denotes the vicinity of the fixed point where the blocking relation is linearizable.