Gluon self-interaction in the position space in Landau gauge

TL;DR

This paper develops a method for analyzing the three-gluon self-interaction vertex in position space within Landau gauge, enabling precise two-loop calculations in SU(N) Yang-Mills theory.

Contribution

It introduces a novel approach to treat the three-gluon vertex in position space, facilitating two-loop calculations with a decomposition into manageable integrals.

Findings

Successfully calculated a two-loop contribution to the Lcc vertex.

Represented integrals as sums of finite and singular contributions.

Applied the uniqueness technique to simplify integrals in position space.

Abstract

We propose a method to treat the three-gluon self-interaction vertex in the position space in D = 4 - 2e dimensions. As an example, we calculate a two-loop contribution to auxiliary Lcc vertex in the Landau gauge which contains the three-gluon vertex for SU(N) Yang-Mills theory. We represent the integral expression as a sum of separate contributions so that each of the contributions is a double finite integral or single integral (singular or finite) in the position space. In each double finite integral we use the freedom to shift exponents in powers in the denominator of integrands by some multiples of e, in order to perform at least one of the integrations by the uniqueness technique without corrupting the first term of the decomposition in e.