Two mutually consistent gauge-invariant decompositions of the nucleon spin were derived by Cho, Ge, and Zhang (2012) using the Cho–Duan–Ge (CDG) Abelian decomposition of the gluon field, which separates 'binding' gluons (the restricted, Abelian part of the gauge field tied to the color direction) from 'valence' gluons (the non-Abelian remainder). The first decomposition attributes nucleon spin to quark spin and orbital angular momentum plus the angular momentum of all gluons; the second includes only binding gluons, excluding valence gluons entirely. The authors argued the second is correct — which would reconfirm the quark-model picture — but conceded that experiments must ultimately decide. The question remains open: the promised SU(3) generalization was never published; the mainstream recasts the controversy as kinetic-versus-canonical decomposition and treats it as largely interpretational; the gluon orbital-angular-momentum sector remains unmeasured; and existing measurement proposals (diffractive and exclusive dijet asymmetries at the Electron-Ion Collider) target the standard decomposition, not the binding/valence split. No observable or lattice-implementable operator that isolates the spin contribution of binding gluons from that of valence gluons has been identified; the closest proposed discriminants (refs. 2 and 3) concern the gluon momentum fraction and the nucleon's constituent content, not the spin split. The surviving open core is to identify observables that separately measure the binding-gluon and valence-gluon contributions in physical SU(3) QCD and thereby decide which decomposition is realized.

Background

The nucleon (proton or neutron) has spin 1/2, and quantum chromodynamics (QCD) must account for it as the sum of the spins and orbital angular momenta (OAM) of its quark and gluon constituents. Deep-inelastic scattering (DIS) experiments in the late 1980s showed that the quark spins carry only a small fraction of the nucleon spin — the 'spin crisis' — so gluon spin and OAM terms must make up the rest, and writing the spin sum rule in a gauge-invariant way became a foundational problem. Two families of gauge-invariant decompositions are now standard: the kinetic (or mechanical) decomposition, in which quark and gluon OAM are defined through covariant derivatives and are measurable through generalized parton distributions, and the canonical decomposition, whose gluon-OAM term is not manifestly gauge invariant and requires a gauge-invariant extension. Reviews by Leader and Lorcé (2014) and Wakamatsu (2014) map this long-standing controversy; a widespread mainstream position is that the differences between decompositions are partly interpretational, corresponding to different bookkeeping of the angular momentum carried by the color fields. Against this backdrop, Cho, Ge, and Zhang (Mod. Phys. Lett. A 27, 1230032 (2012)) applied the Cho–Duan–Ge (CDG) Abelian decomposition, which at each spacetime point splits the SU(2) gauge potential into a 'restricted' Abelian part built from a color direction field n^\hat{n} (the binding gluons) and a residual non-Abelian part (the valence gluons). From this they built two gauge-invariant nucleon-spin decompositions: decomposition (12), in which the gluon angular momentum includes contributions from all gluons (binding plus valence), and decomposition (20), in which only binding gluons contribute and valence gluons are excluded. They argued that (20) is the physically correct one — in which case valence gluons carry no nucleon spin and the quark model is reconfirmed — while acknowledging that if (12) were correct, 'the quark model will be seriously challenged.' The paper states that the SU(2) construction generalizes to SU(3) QCD (the physical three-color theory) in a 'non-trivial, but straightforward' way to be presented elsewhere, and defers the associated problem of gauge-invariant quark/gluon momentum operators to a companion paper. The two decompositions make physically different statements about how much of the nucleon spin the gluon sector carries, so the question of which is realized is, in the authors' formulation, an empirical one — but no observable that measures the binding-gluon contribution separately from the valence-gluon contribution was proposed, and none has been identified since. The Electron-Ion Collider (EIC), a planned electron–proton collider designed to map gluon structure, is the natural facility where any such discriminating measurement would be made.

Problem Statement

For the nucleon, two mutually consistent gauge-invariant decompositions of the total spin have been derived from the Cho–Duan–Ge Abelian decomposition of the gluon field, which splits the color gauge potential into 'binding' gluons (the restricted, Abelian part of the gauge field tied to the color-direction field) and 'valence' gluons (the residual non-Abelian part): one decomposition in which the gluon angular momentum is carried by all gluons (binding plus valence), and one in which only binding gluons contribute and valence gluons are excluded. Which decomposition is physically realized, and how can this be decided? Concretely, the problem is to identify how the binding-gluon and valence-gluon contributions to the nucleon spin can be measured separately in physical SU(3) QCD — the three-color gauge theory of the strong interaction, with dynamical quarks — for example through a factorization theorem relating the corresponding gauge-invariant operators to a measurable cross-section asymmetry, or through operators implementable in lattice QCD (the discrete numerical formulation of the theory). The task is resolved by exhibiting such a discriminating observable, operator, or no-go theorem; determining which decomposition is physically realized — and hence whether valence gluons contribute to the nucleon spin — is the subsequent experimental step and is not itself part of the deliverable. Only the SU(2) version of the two decompositions was worked out in the source; the generalization to SU(3) was promised there but never published.

References

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