Nucleon Spin DecompositionEdit

Nucleon spin decomposition addresses how the proton’s intrinsic angular momentum of 1/2 emerges from its quark and gluon constituents within the framework of quantum chromodynamics (QCD). The problem grew out of surprising experimental results in the late 1980s showing that the spins of the up and down quarks contribute only a fraction of the proton’s total spin. Since then, theorists have devised several ways to partition the proton’s spin into pieces associated with quark spin, quark orbital motion, and gluon contributions. The task is not only to write down a consistent mathematical decomposition but also to connect those pieces to quantities that can be accessed, in principle, by experiments or lattice simulations. The debate reflects deep questions about gauge invariance, the interpretation of parton dynamics, and how to extract meaningful physical content from a highly interacting quantum field theory.

Two main lines of theoretical development define the landscape. One emphasizes gauge-invariant, observable quantities that can be tied to measurable distributions. The other emphasizes the intuitive picture suggested by the parton model, which is natural in high-energy scattering but relies on choices of gauge and frame. Both perspectives agree on the basic fact that the nucleon’s spin must be shared among its constituents, but they disagree on the most natural and most practically measurable way to assign portions to quark spin, quark orbital angular momentum, and gluon angular momentum. The current picture rests on a combination of experimental measurements, generalized parton distributions, and lattice computations, with ongoing work to sharpen the separation between different components.

Theoretical frameworks

Ji decomposition

The Ji framework provides a gauge-invariant way to express the total spin of the nucleon as a sum of quark and gluon contributions that are themselves well defined operators in QCD. In this scheme, the total angular momentum carried by quarks is J_q^Ji, and the total angular momentum carried by gluons is J_g^Ji, with the quark spin ΔΣ and the quark orbital piece L_q^Ji arranged so that

nucleon spin = ΔΣ/2 + L_q^Ji + J_g^Ji.

A central feature is the Ji sum rule, which links the total quark angular momentum to moments of generalized parton distributions (GPDs). Specifically, for each quark flavor q, the total quark angular momentum is given by

J_q^Ji = 1/2 ∫_0^1 dx x [H_q(x, ξ, t=0) + E_q(x, ξ, t=0)],

and a similar relation holds for gluons with the gluon GPDs H_g and E_g. Here, H and E are functions encoding the distribution of partons inside the nucleon with respect to both momentum and spatial information, and ξ is a skewness parameter that vanishes in the forward limit. The upshot is that Ji’s decomposition pins the spin budget to gauge-invariant observables that can be accessed through processes such as deeply virtual Compton scattering Deeply virtual Compton scattering and related exclusive reactions Generalized parton distributions.

ΔΣ denotes the quark helicity sum, obtainable from polarized deep inelastic scattering Deep inelastic scattering;
L_q^Ji is the quark orbital angular momentum component, which is not directly measurable as a standalone quantity but is inferred from the remainder after subtracting ΔΣ/2 from J_q^Ji;
J_g^Ji is the total gluon angular momentum, combining gluon spin and gluon orbital motion in a gauge-invariant way.

Jaffe-Manohar decomposition

The Jaffe-Manohar framework presents a more canonical split that aligns with the parton-model intuition. It writes the nucleon spin as

nucleon spin = S_q + L_q^can + S_g + L_g^can,

where S_q = ΔΣ/2 is the quark spin, L_q^can is the quark canonical orbital angular momentum, S_g is the gluon spin, and L_g^can is the gluon canonical orbital angular momentum. This decomposition mirrors the familiar separation in nonrelativistic quantum mechanics, but its individual pieces are not all gauge invariant in general. It is most transparent in light-cone gauge and in formulations that emphasize parton kinematics, and there are ongoing efforts to relate its components to gauge-invariant and process-dependent observables. The canonical pieces can be linked conceptually to Wigner distributions, which encode joint position–momentum correlations, but extracting a clean, universal experimental separation of L_q^can and L_g^can remains a challenge.

Gauge invariance and frame dependence

A core issue in nucleon spin decomposition is the tension between a decomposition that is mathematically clean and one that reflects what can be measured in experiments. Ji’s decomposition is built to be gauge invariant and is naturally connected to the second moments of GPDs, making it robust against the choice of gauge. The Jaffe-Manohar terms offer a more intuitive picture in the parton model but are not gauge invariant at the level of individual components, which raises questions about their direct observability outside special gauges and definitions. In practice, most experimental programs aim to determine J_q^Ji and J_g^Ji (and therefore infer ΔΣ and the remainder as orbital and gluon angular momentum) rather than trying to isolate S_q^can or L_q^can separately in an unambiguous, gauge-independent way. The evolution of these quantities with the energy scale Q^2, governed by QCD, adds another layer of complexity, as the relative importance of quark and gluon contributions shifts with scale.

Experimental status and lattice results

Experimental data on quark and gluon spin

Polarized deep inelastic scattering has established that the quark helicity contribution ΔΣ to the proton spin is nonzero but smaller than naïve expectations, typically in the range of about 0.25 to 0.35 at accessible Q^2 values. This finding is a central element of the so-called spin puzzle. Direct access to the total angular momentum carried by quarks (J_q^Ji) and gluons (J_g^Ji) comes from processes sensitive to GPDs, such as deeply virtual Compton scattering, and from global analyses that combine various exclusive and semi-inclusive measurements. Polarized proton–proton collisions (e.g., at RHIC) have constrained the gluon polarization ΔG, with results suggesting a non-negligible but not yet precisely pinned-down contribution over the probed range of parton momentum fractions x. The overall picture is that quarks provide a substantial but incomplete share, with a sizable role played by orbital motion and gluon dynamics.

Lattice QCD insights

Lattice QCD simulations have matured to the point where they can compute the second moments of GPDs and extract Ji’s J_q^Ji and J_g^Ji with controlled uncertainties. Across several calculations, the total quark angular momentum J_q^Ji tends to lie in a broad corridor that supports a substantial quark contribution but leaves room for large orbital components. The gluon angular momentum J_g^Ji is harder to pin down with precision but is found to be nonzero and important in the total budget. While exact numbers vary with lattice setup, quark and gluon contributions together account for the bulk of the proton’s spin, with the remainder typically interpreted as orbital motion (and its scale dependence) plus any subtle gluon effects that are hardest to disentangle experimentally.

Controversies and debates

Gauge-invariant versus canonical decompositions: The Ji decomposition is gauge invariant and closely tied to GPD moments, which makes it appealing for connecting theory to experiment. The Jaffe-Manohar (canonical) decomposition mirrors the intuitive parton picture but requires specific gauge choices to define its components uniquely, leading to debates about what can be claimed as a physical, observable piece of angular momentum in a gauge theory.
Observability of orbital components: Extracting the separate orbital angular momentum pieces, especially L_q^Ji vs L_q^can and L_g^can, is challenging. While J_q^Ji and J_g^Ji can be accessed (at least in principle) through GPDs and related observables, the direct separation of orbital pieces from spin is more model-dependent and subject to ongoing refinement.
Spin sum rules and scale evolution: The decomposition depends on the energy scale Q^2, and the partition of spin among quark and gluon sectors evolves under QCD. This scale dependence complicates comparisons across experiments performed at different kinematic regimes and calls for careful treatment in global analyses.
Axial anomaly and the sea: The quark spin ΔΣ is sensitive to the axial anomaly, which affects how spin is redistributed among quark flavors and gluons as one varies Q^2. The interpretation of this redistribution is part of the broader discourse on how much of the proton’s spin sits in intrinsic quark spin versus gluon spin and orbital motion.
Response to broader scientific narratives: In discussions about fundamental structure, some observers emphasize that the physics of spin is a clean testbed for gauge theories and hadron structure, while critics argue about broader social narratives shaping research priorities. From the standpoint of a results-driven program, the core interest remains in obtaining precise, testable statements about how spin arises from QCD dynamics, rather than broad interpretive frameworks.

Applications and implications

A practical takeaway is that most of the proton’s spin arises from a combination of quark helicity, gluon angular momentum, and orbital motion, with the precise balance still being refined. This understanding informs models of nucleon structure, guides the interpretation of spin-dependent scattering data, and supports the ongoing development of theoretical tools such as GPDs and Wigner distributions.
Experimental programs and future facilities: The development of high-precision measurements of GPDs and related observables at facilities such as the Electron-Ion Collider (EIC) is motivated by the goal of pinning down the Ji sum rule with greater accuracy, thereby sharpening the picture of how J_q^Ji and J_g^Ji partition the proton spin. These efforts complement lattice QCD studies and deepen our understanding of how QCD binds spin and momentum in a strongly interacting system. For more on the experimental side, see Deeply virtual Compton scattering and Generalized parton distributions.
Broader significance: The spin decomposition story reinforces a broader engineering-minded approach to fundamental physics: focus on well-defined, measurable quantities, test predictions against data, and recognize that multiple internally consistent frameworks can illuminate different facets of a complex system. It also illustrates how progress in one domain (hadron structure) benefits from advances in theory, computation, and experimental technique.