On the decomposition theorem for gluons

TL;DR

This paper proves a decomposition theorem for gluons in non-Abelian gauge theory using the contour gauge, clarifying the conditions under which physical gluon components can be separated from pure gauge parts.

Contribution

It reformulates the gluon decomposition as a theorem and proves it within the contour gauge framework, providing mathematical evidence for the decomposition in non-Abelian theories.

Findings

Gluon decomposition can be rigorously established as a theorem in the contour gauge.

Contour gauge has residual gauge freedoms related to boundary configurations.

Boundary conditions influence the physical gluon component extraction.

Abstract

Recently, the problem of spin and orbital angular momentum (AM) separation has widely been discussed. Nowadays, all discussions about the possibility to separate the spin AM from the orbital AM in the gauge invariant manner are based on the ansatz that the gluon field can be presented in form of the decomposition where the physical gluon components are additive to the pure gauge gluon components, i.e. $A_\mu = A_\mu^{\text{phys}}+A_\mu^{\text{pure}}$. In the present paper, we show that in the non-Abelian gauge theory this gluon decomposition has a strong mathematical evidence in the frame of the contour gauge conception. In other words, we reformulate the gluon decomposition ansatz as a theorem on decomposition and, then, we use the contour gauge to prove this theorem. In the first time, we also demonstrate that the contour gauge possesses the special kind of residual gauge related to the boundary field configurations and expressed in terms of the pure gauge fields. As a result, the trivial boundary conditions lead to the inference that the decomposition includes the physical gluon configurations only provided the contour gauge condition.