Regularization, Renormalization, and Dimensional Analysis: Dimensional Regularization meets Freshman E&M

TL;DR

This paper demonstrates the use of dimensional regularization in elementary electrostatics, comparing it with cutoff regularization, and discusses renormalization schemes and dimensional transmutation in extra dimensions.

Contribution

It introduces dimensional regularization to simple electrostatics problems, highlighting its advantages over cutoff methods and exploring renormalization schemes and dimensional transmutation.

Findings

Dimensional regularization preserves translational symmetry.

Comparison shows advantages over cutoff regularization.

Application to extra dimensions illustrates dimensional transmutation.

Abstract

We illustrate the dimensional regularization technique using a simple problem from elementary electrostatics. We contrast this approach with the cutoff regularization approach, and demonstrate that dimensional regularization preserves the translational symmetry. We then introduce a Minimal Subtraction (MS) and a Modified Minimal Subtraction (MS-Bar) scheme to renormalize the result. Finally, we consider dimensional transmutation as encountered in the case of compact extra-dimensions.