Computing the Effective Action with the Functional Renormalization Group

TL;DR

This paper discusses how to compute the effective action using the functional renormalization group, providing examples in various theories including scalar, gauge, and gravity, and reproducing known results like scattering amplitudes and vacuum polarization.

Contribution

It demonstrates practical calculations of the effective action via the functional renormalization group in different field theories, including non-local heat kernel techniques.

Findings

Reproduces four-point scattering amplitudes in scalar and pion theories

Calculates vacuum polarization in QED and Yang-Mills theories

Derives two-point functions for scalars and gravitons in gravity theories

Abstract

The "exact" or "functional" renormalization group equation describes the renormalization group flow of the effective average action $\Gamma_k$. The ordinary effective action $\Gamma_0$ can be obtained by integrating the flow equation from an ultraviolet scale $k=\Lambda$ down to $k=0$. We give several examples of such calculations at one-loop, both in renormalizable and in effective field theories. We use the results of Barvinsky, Vilkovisky and Avramidi on the non-local heat kernel coefficients to reproduce the four point scattering amplitude in the case of a real scalar field theory with quartic potential and in the case of the pion chiral lagrangian. In the case of gauge theories, we reproduce the vacuum polarization of QED and of Yang-Mills theory. We also compute the two point functions for scalars and gravitons in the effective field theory of scalar fields minimally coupled to gravity.