The Abelian (Cho–Duan–Ge) decomposition splits the gluon field of QCD gauge-independently into a color-neutral 'binding' component (the restricted field, which carries the non-Abelian monopole background) and a color-carrying 'valence' component (the chromon). Using this split, Cho, Ge, and Zhang constructed two gauge-invariant decompositions of the nucleon spin — one counting all gluons, one counting only the binding gluons — and posed a sharp physical question: are valence gluons constituents of the nucleon at all, or only of glueballs, and hence which decomposition of nucleon spin (and, by the companion analysis, of nucleon momentum, with gluon momentum fractions predicted at roughly 51%, 21%, or 6% depending on gluon content and on which momentum operator is measurable) is physically realized? More than a decade later the question is unresolved. The spin-decomposition literature debates the uniqueness and observability of gauge-invariant decompositions without settling the two-gluon question; lattice QCD quotes only total glue contributions; and the concrete experimental discriminants proposed in 2018–2023 — gluon jets composed 1:3 of sharp, low-multiplicity 'neuron' jets (color factor 3/4) and broad 'chromon' jets (color factor 9/4) with distinct color-flow patterns — remain untested, while a 2024 re-interpretation of ALEPH/DELPHI and CMS data as circumstantial evidence for two gluon types awaits independent confirmation. The open problem is to establish whether a physically meaningful, gauge- and scheme-independent separation of the nucleon's gluonic content into binding and valence components exists, whether the experimentally measured gluon momentum and gluonic angular momentum include the valence-gluon contribution, and how this can be probed experimentally.

Background

The proton and neutron (collectively, the nucleon) have spin 1/2, and polarized deep-inelastic scattering measurements in the late 1980s revealed that the spins of the quarks carry much less than half of it — the origin of the long-standing 'nucleon spin crisis'. Accounting for the nucleon's spin and momentum in terms of its quark and gluon constituents requires decomposing them into quark spin, quark orbital angular momentum (OAM, the angular momentum of a constituent's motion about the nucleon center), gluon spin, and gluon OAM. The canonical decomposition of Jaffe and Manohar is physically transparent but gauge-dependent (its terms depend on the choice of gauge potential), while the gauge-invariant decomposition of Ji splits the nucleon spin into quark spin, quark OAM, and total gluon angular momentum without separating the gluon piece into spin and OAM. A 2008 claim that a fully gauge-invariant canonical decomposition exists triggered a controversy over whether, and in which sense, the photon or gluon angular momentum can be split into spin and orbital parts in a physically meaningful way.

A distinctive entry in this debate is the Abelian decomposition (also called the Cho–Duan–Ge decomposition). One chooses a color direction field n^(x)\hat{n}(x) in internal color space and imposes the magnetic symmetry condition Dμn^=0D_\mu\hat{n}=0; the gluon potential then splits uniquely into a 'restricted' potential A^μ\hat{A}_\mu that keeps the full gauge degrees of freedom and carries a topological non-Abelian monopole background, and a remainder XμX_\mu that transforms covariantly like a colored matter field. The split is gauge-independent once n^\hat{n} is fixed. In this framework the quanta of A^μ\hat{A}_\mu are called binding gluons (later renamed 'neurons'), argued to be responsible for confinement of colored sources, while the quanta of XμX_\mu are called valence gluons (later renamed 'chromons'): colored objects that, like quarks, must themselves be confined, forming glueballs (color-singlet bound states of gluons) rather than residing in ordinary hadrons.

Cho, Ge, and Zhang used this decomposition to write down two gauge-invariant decompositions of the nucleon spin, both satisfying the angular-momentum commutation algebra: one in which all gluons (binding plus valence) contribute, and one in which only the binding gluons contribute. They argued from the quark model — low-lying nucleons as bound states of three valence quarks with no valence gluons — that the second should be physically correct, but stated explicitly that 'ultimately experiments should determine which is correct', noting that endorsement of the first would seriously challenge the quark model. The companion momentum analysis by Cho, Ge, Pak, and Zhang predicts that the fraction of nucleon momentum carried by gluons is about 51% if all gluons contribute and the kinematic momentum is measurable, about 21% if only binding gluons contribute (or, in a different measurability assumption, if all gluons contribute but only the canonical momentum is measurable), or about 6% if only binding gluons contribute under the canonical reading — so that, in the authors' words, measuring the gluon momentum in nucleons would tell 'which gluons are in nucleons'. Which of these competing momentum and angular-momentum operators corresponds to what experiments actually measure is itself part of what remains unresolved.

The broader spin-decomposition community has not adopted the binding/valence gluon distinction. The comprehensive review by Leader and Lorcé classifies the Cho decompositions as gauge-invariant extensions obtained with a Lorenz-type constraint (a gauge-condition-like subsidiary condition imposed on the decomposition), members of an infinite family of mathematically consistent decompositions, and argues that much of the controversy is interpretational. Wakamatsu analyzed the uniqueness and observability of such decompositions, arguing that not all of them can represent independent physics, but did not address the two-gluon question. Lattice QCD calculations of the quark and glue spin and angular momentum in the nucleon quote only the total glue contributions. Meanwhile the Cho program has developed concrete experimental discriminants: since the two gluon types carry different color charges, gluon jets at colliders should be a 1:3 mixture of narrow, low-multiplicity neuron jets with effective color factor 3/4 and broader-than-quark-jet chromon jets with color factor 9/4, with distinct color-flow (dipole) patterns; and a 2024 derivation of coupled evolution equations for the neuron and chromon distributions found a robust 3:1 chromon-to-neuron ratio at small momentum fraction. Here 'color factor' means the group-theoretical factor that weights the radiation intensity of a parton according to its color representation. No independent experimental analysis has tested these predictions.

Problem Statement

Determine the gluonic constituent structure of the nucleon as posed by the Abelian decomposition of QCD. Specifically: (i) Does the gauge-independent split of the gluon field into the restricted (binding) potential A^μ\hat{A}_\mu and the valence potential XμX_\mu define physically meaningful, separately observable contributions of binding gluons and valence gluons (chromons) to the nucleon — in particular, is this separation independent of the choice of the color direction field n^\hat{n} and stable under renormalization and evolution — or is it an unphysical rewriting with no observable consequence? (ii) In consequence, do the experimentally measured gluon momentum fraction and the gluonic angular-momentum contribution of the nucleon include the valence-gluon piece, i.e. which of the competing gauge-invariant decompositions of nucleon spin and momentum (all-gluon versus binding-gluon-only, with the associated predicted gluon momentum fractions of roughly 51%, 21%, or 6%) is physically realized? (iii) How can the binding-versus-valence gluon content of the nucleon be probed experimentally — for example through the predicted two-component structure of gluon jets (a 1:3 neuron-to-chromon mixture with color factors 3/4 and 9/4, distinct jet-width, multiplicity, and color-flow signatures) or through any other observable that discriminates the two-gluon hypothesis from standard single-gluon QCD — so that the question of which gluons constitute the nucleon is settled by theory or experiment? (i) is the primary deliverable, and its two components — independence of the color direction field n^\hat{n} and stability under renormalization and evolution — may be resolved or refuted separately; a negative answer to (i) resolves the problem by establishing the split as physically vacuous and renders (ii) moot, whereas (ii) and (iii) are the operative deliverables only when (i) is answered affirmatively, and (iii) alone — proposing a discriminating observable — does not by itself establish which decomposition is physically realized.

References

  1. Y. M. Cho, Mo-Lin Ge, Pengming Zhang, 'Nucleon Spin in QCD: Old Crisis and New Resolution', Modern Physics Letters A 27, 1230032 (2012). DOI: 10.1142/S0217732312300327; arXiv:1010.1080; https://arxiv.org/abs/1010.1080
  2. Y. M. Cho, Mo-Lin Ge, D. G. Pak, Pengming Zhang, 'Nucleon Momentum Decomposition in QCD', arXiv:1102.1130 (2011); https://arxiv.org/abs/1102.1130
  3. Xiangdong Ji, 'Gauge-Invariant Decomposition of Nucleon Spin', Physical Review Letters 78, 610–613 (1997). DOI: 10.1103/PhysRevLett.78.610; https://doi.org/10.1103/PhysRevLett.78.610
  4. R. L. Jaffe, A. Manohar, 'The g1 problem: Deep inelastic electron scattering and the spin of the proton', Nuclear Physics B 337, 509–546 (1990). DOI: 10.1016/0550-3213(90)90506-9; https://doi.org/10.1016/0550-3213(90)90506-9
  5. Masashi Wakamatsu, 'Are there infinitely many decompositions of the nucleon spin?', Physical Review D 87, 094035 (2013). DOI: 10.1103/PhysRevD.87.094035; arXiv:1302.5152; https://arxiv.org/abs/1302.5152
  6. Masashi Wakamatsu, 'Is gauge-invariant complete decomposition of the nucleon spin possible?', International Journal of Modern Physics A 29, 1430012 (2014). DOI: 10.1142/S0217751X14300129; arXiv:1402.4193; https://arxiv.org/abs/1402.4193
  7. Elliott Leader, Cédric Lorcé, 'The angular momentum controversy: What's it all about and does it matter?', Physics Reports 541, 163–248 (2014). DOI: 10.1016/j.physrep.2014.02.010; arXiv:1309.4235; https://arxiv.org/abs/1309.4235
  8. D. G. Pak, P. M. Zhang, 'Consistent gauge invariant nucleon spin decomposition', arXiv:1204.5030 (2012); https://arxiv.org/abs/1204.5030
  9. Keh-Fei Liu, 'Quark and Glue Components of the Proton Spin from Lattice Calculation', International Journal of Modern Physics: Conference Series 40, 1660005 (2016). DOI: 10.1142/S2010194516600053; arXiv:1504.06601; https://arxiv.org/abs/1504.06601
  10. Y. M. Cho, Franklin H. Cho, Pengming Zhang, 'Probing Two types of Gluon Jets at LHC', arXiv:1812.11300 (2018); https://arxiv.org/abs/1812.11300
  11. Y. M. Cho, Pengming Zhang, Li-Ping Zou, 'Experimental verification of two types of gluon jets in QCD', Physical Review D 107, 054024 (2023). DOI: 10.1103/PhysRevD.107.054024; arXiv:2212.14817; https://arxiv.org/abs/2212.14817
  12. Yirui Yang, Wei Kou, Xiaopeng Wang, Yanbing Cai, Xurong Chen, 'Deriving the QCD evolution equations under the Abelian decomposition scheme', European Physical Journal C 84, 924 (2024). DOI: 10.1140/epjc/s10052-024-13280-w; arXiv:2401.11442; https://arxiv.org/abs/2401.11442
  13. Y. M. Cho, 'Two Types of Gluons in QCD: Re-interpretation of ALEPH and CMS Gluon Jet Data', arXiv:2401.08704 (2024); https://arxiv.org/abs/2401.08704