On the controversy concerning the definition of quark and gluon angular momentum

TL;DR

This paper addresses the controversy in QCD regarding the proper gauge-invariant decomposition of total angular momentum into quark and gluon parts, clarifying the validity of various proposed operators and the physical meaning of gluon spin and orbital contributions.

Contribution

It advocates for the canonical operators as the physically meaningful choice and clarifies the gauge invariance limitations of splitting gluon angular momentum into spin and orbital parts.

Findings

Canonical operators are the correct physical choice.

Gluon angular momentum cannot generally be split gauge-invariantly into spin and orbital parts.

Gluon helicity is gauge invariant and measurable in deep inelastic scattering.

Abstract

A major controversy has arisen in QCD as to how to split the total angular momentum into separate quark and gluon contributions, and as to whether the gluon angular momentum can itself be split, in a gauge invariant way, into a spin and orbital part. Several authors have proposed various answers to these questions and offered a variety of different expressions for the relevant operators. I argue that none of these is acceptable and suggest that the canonical expression for the momentum and angular momentum operators is the correct and physically meaningful one. It is then an inescapable fact that the gluon angular momentum operator cannot, in general, be split in a gauge invariant way into a spin and orbital part. However, the projection of the gluon spin onto its direction of motion i.e. its helicity is gauge invariant and is measured in deep inelastic scattering on nucleons. The Ji sum rule, relating the quark angular momentum to generalized parton distributions, though not based on the canonical operators, is shown to be correct, if interpreted with due care. I also draw attention to several interesting aspects of QED and QCD, which, to the best of my knowledge, are not commented upon in the standard textbooks on Field Theory.