Are there infinitely many decompositions of the nucleon spin ?

TL;DR

This paper examines the non-uniqueness of nucleon spin decomposition by analyzing gauge invariance and path dependence, providing evidence that the number of physically distinct decompositions is finite, contrary to some recent claims.

Contribution

It demonstrates gauge- and path-independence of the nucleon spin decomposition within a specific class of gauges, challenging the idea of infinitely many decompositions.

Findings

Gauge-independence of the 1-loop evolution matrix

Path-independence within a class of gauges

Arguments against infinite decompositions of nucleon spin

Abstract

We discuss the uniqueness or non-uniqueness problem of the decomposition of the gluon field into the physical and pure-gauge components, which is the basis of the recently proposed two physically inequivalent gauge-invariant decompositions of the nucleon spin. It is crucialy important to recognize the fact that the standard gauge fixing procedure is essentially a process of projecting out the physical components of the massless gauge field. A complexity of the nonabelian gauge theory as compared with the abelian case is that a closed expression for the physical component can be given only with use of the non-local Wilson line, which is generally path-dependent. It is known that, by choosing an infinitely long straight-line path in space and time, the direction of which is characterized by a constant 4-vector $n^\mu$, one can cover a class of gauge called the general axial gauge, containing three popular gauges, i.e. the temporal, the light-cone, and the spatial axial gauge. Within this general axial gauge, we have calculated the 1-loop evolution matrix for the quark and gluon longitudinal spins in the nucleon. We found that the final answer is exactly the same independently of the choices of $n^\mu$, which amounts to proving the gauge-independence and path-independence simultaneously, although within a restricted class of gauges and paths. By drawing on all of these findings together with well-established knowledge from gauge theories, we argue against the rapidly spreading view in the community that there are infinitely many decompositions of the nucleon spin.