Point-splitting regularization of composite operators and anomalies

TL;DR

This paper reviews the point-splitting regularization method for composite operators, focusing on anomaly calculations, and introduces algebraic tools and alternative approaches to simplify the process.

Contribution

It provides a detailed pedagogical review and develops algebraic tools for handling point-splitting regularization and anomalies, offering an alternative to standard methods.

Findings

Unified treatment of various anomalies using point-splitting

Introduction of algebraic tools for path-ordered exponentials

Simplified calculation via deformed point-split transformations

Abstract

The point-splitting regularization technique for composite operators is discussed in connection with anomaly calculation. We present a pedagogical and self-contained review of the topic with an emphasis on the technical details. We also develop simple algebraic tools to handle the path ordered exponential insertions used within the covariant and non-covariant version of the point-splitting method. The method is then applied to the calculation of the chiral, vector, trace, translation and Lorentz anomalies within diverse versions of the point-splitting regularization and a connection between the results is described. As an alternative to the standard approach we use the idea of deformed point-split transformation and corresponding Ward-Takahashi identities rather than an application of the equation of motion, which seems to save the complexity of the calculations.