Contour gauge: Compendium of Results in Theory and Applications

TL;DR

This review explores the contour gauge, a non-local gauge in QCD, highlighting its advantages, the role of additional diagram contributions, and its implications for gauge invariance, angular momentum separation, and boundary conditions.

Contribution

It provides a comprehensive analysis of the contour gauge, demonstrating its benefits over local gauges and its impact on gauge invariance, diagram contributions, and gluon decomposition in QCD.

Findings

Contour gauge eliminates residual gauge ambiguities.

Additional diagrams restore gauge invariance of the hadron tensor.

Boundary conditions relate to pure gauge fields.

Abstract

In this review, we outline the main features of the non-local gauge, named the contour gauge. The contour gauge belongs to the axial type of gauges and extends the local gauge used in the most of approaches. The geometry of gluon fields and the path-dependent formalism are the essential tools for the description of non-local gauges. The principle feature of the contour gauge is that there are no the residual gauges which are left in the finite domain of space. In the review, we present the useful correspondence between the contour gauge conception and the Hamiltonian (Lagrangian) formalism. The Hamiltonian formalism is turned out to be a very convenient framework for the understanding of contour gauges. The comprehensive comparison analysis of the local and non-local gauges advocates the advantage of the contour gauge use. We show that the appropriate use the contour gauge leads to the existence of extra diagram contributions. These additional contributions, first, restore the gauge invariance of the hadron tensor and, second, give the important terms for the observable quantities. We also demonstrate the significant role of the additional diagrams to form the relevant contour in the Wilson path-ordered exponential. Ultimately, it leads to the spurious singularity fixing. Moreover, in the present review, we discuss in detail the problem of spin and orbital angular momentum separation. We show that in $SU(3)$ gauge theories the gluon decomposition on the physical and pure gauge components has a strong mathematical evidence provided the contour gauge conception has been used. In addition, we prove that the contour gauge possesses the special kind of residual gauge that manifests at the boundary of space. Besides, the boundary field configurations can be associated with the pure gauge fields.