Practicalities of renormalizing quantum field theories

TL;DR

This paper reviews practical techniques for renormalizing quantum field theories, focusing on multi-loop calculations, algorithm implementation, and applications to phi^4 theory, QCD, and non-local operators.

Contribution

It provides a systematic overview of renormalization methods, including computer algebra implementation and applications to complex theories like QCD and non-local operators.

Findings

Renormalization techniques for multi-loop quantum field theories.

Implementation of algorithms within computer algebra systems.

Two-loop correction to the Gribov mass gap equation in QCD.

Abstract

We review the techniques used to renormalize quantum field theories at several loop orders. This includes the techniques to systematically extract the infinities in a Feynman integral and the implementation of the algorithm within computer algebra. To illustrate the method we discuss the renormalization of phi^4 theory and QCD including the application of the critical point large $N$ technique as a check on the anomalous dimensions. The renormalization of non-local operators in QCD is also discussed including the derivation of the two loop correction to the Gribov mass gap equation in the Landau gauge.