The Three-point Function in Split Dimensional Regularization in the Coulomb Gauge

TL;DR

This paper introduces split dimensional regularization, a gauge-invariant method for evaluating Coulomb gauge integrals in non-Abelian theories, ensuring well-defined results and consistent BRST identities.

Contribution

The paper develops and applies split dimensional regularization to Coulomb gauge integrals, including nonlocal and multi-propagator cases, and discusses modifications to BRST identities.

Findings

Split dimensional regularization provides a well-defined framework for Coulomb gauge integrals.

Both quark self-energy and vertex functions remain local despite nonlocal integrals.

The standard BRST identity is extended to include ghost contributions.

Abstract

We use a gauge-invariant regularization procedure, called ``split dimensional regularization'', to evaluate the quark self-energy $\Sigma (p)$ and quark-quark-gluon vertex function $\Lambda_\mu (p^\prime,p)$ in the Coulomb gauge, $\vec{\bigtriangledown}\cdot\vec{A}^a = 0$. The technique of split dimensional regularization was designed to regulate Coulomb-gauge Feynman integrals in non-Abelian theories. The technique which is based on two complex regulating parameters, $\omega$ and $\sigma$, is shown to generate a well-defined set of Coulomb-gauge integrals. A major component of this project deals with the evaluation of four-propagator and five-propagator Coulomb integrals, some of which are nonlocal. It is further argued that the standard one-loop BRST identity relating $\Sigma$ and $\Lambda_\mu$, should by rights be replaced by a more general BRST identity which contains two additional contributions from ghost vertex diagrams. Despite the appearance of nonlocal Coulomb integrals, both $\Sigma$ and $\Lambda_\mu$ are local functions which satisfy the appropriate BRST identity. Application of split dimensional regularization to two-loop energy integrals is briefly discussed.