Lectures on the functional renormalization group method

TL;DR

This paper provides an overview of the functional renormalization group method, its theoretical foundations, and diverse applications including quantum field theories, scalar models, and gauge theories, highlighting its versatility beyond critical phenomena.

Contribution

It introduces various formulations of the renormalization group equations, including non-perturbative and gauge-invariant approaches, and discusses their applications to different physical models.

Findings

Resummation of loop-expansion via Wegner-Houghton equation

Demonstration of flattening of the effective potential in sine-Gordon model

Development of gauge-invariant evolution equations for QED

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.