Introduction to the technique of dimensional regularization

Abstract

The purpose of this review article is to explain and illustrate in detail the technique of dimensional regularization, which is a major mathematical tool in the renormalization program of gauge theories. The most important single feature of the new technique is the concept of analytic continuation in the number of space-time dimensions $2\ensuremath{\omega}$, where the regulating parameter $\ensuremath{\omega}$ is complex in general, and $\ensuremath{\omega}=2$ corresponds to four-dimensional space-time. The technique of dimensional regularization preserves the local gauge symmetry of the underlying Lagrangian and thereby permits a consistent gauge-invariant treatment of divergent Feynman integrals to all orders in perturbation theory. The method can thus be applied---as demonstrated in this article---not only to Abelian gauge models, but more importantly to non-Abelian theories such as Yang-Mills fields and quantum gravity, to which the majority of conventional regularization procedures is inapplicable. We illustrate both the advantages and the limitation of dimensional regularization, as well as its extension to massless particles.