First results for the Coulomb gauge integrals using NDIM

TL;DR

This paper applies the negative dimensional integration method to Coulomb gauge integrals, providing complete finite and divergent results at one- and two-loop levels, addressing issues with energy integrals in perturbative calculations.

Contribution

It introduces a novel application of NDIM with split-dimension parameters to Coulomb gauge integrals, offering comprehensive results for arbitrary propagator exponents.

Findings

Complete finite and divergent parts of integrals obtained

Results valid for arbitrary exponents and dimensions

Addresses energy integral issues in Coulomb gauge perturbation

Abstract

The Coulomb gauge has at least two advantadges over other gauge choices in that bound states between quarks and studies of confinement are easier to understand in this gauge. However, perturbative calculations, namely Feynman loop integrations are not well-defined (there are the so-called energy integrals) even within the context of dimensional regularization. Leibbrandt and Williams proposed a possible cure to such a problem by splitting the space-time dimension into $D=\omega+\rho$, i.e., introducing a specific one parameter $\rho$ to regulate the energy integrals. The aim of our work is to apply negative dimensional integration method (NDIM) to the Coulomb gauge integrals using the recipe of split-dimension parameters and present complete results -- finite and divergent parts -- to the one and two-loop level for arbitrary exponents of propagators and dimension.