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

Abstract

We use a gauge-invariant regularization procedure, called ``split dimensional regularization'', to evaluate the quark self-energy $Σ(p)$ and quark-quark-gluon vertex function $Λ_μ(p^\prime,p)$ 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, $ω$ and $σ$, 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 $Σ$ and $Λ_μ$, 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 $Σ$ and $Λ_μ$ are local functions which satisfy the appropriate BRST identity. Application of split dimensional regularization to two-loop energy integrals is briefly discussed.