Spin StructureEdit

Spin structure is the study of how the intrinsic angular momentum of hadrons, especially the nucleon, is built from the spins and orbital motion of their constituent quarks and gluons within the framework of Quantum Chromodynamics (QCD). The proton and neutron each carry spin 1/2, but the way this spin emerges from the underlying quark-gluon dynamics has been a central question in nuclear and particle physics for decades. Experimental probes that use polarized beams and targets extract spin-dependent observables, notably the spin structure functions g1 and g2, which encode information about how partons—quarks and gluons—carry spin and orbital angular momentum. The story begins with surprising results in the late 1980s and continues with ongoing efforts to map out the complete decomposition of spin in terms of quark spins, gluon spins, and orbital angular momentum.

A principal goal in the field is to decompose the nucleon spin into its fundamental components and to understand how these components depend on resolution scales set by QCD. The basic relation is that the total spin J of the nucleon (J = 1/2) can be written schematically as a sum over contributions from quark spins (ΔΣ), gluon spin (ΔG), and orbital angular momentum of quarks and gluons (L_q and L_g). In symbols, 1/2 = 1/2 ΔΣ + ΔG + L_q + L_g. The quark-spin part ΔΣ is itself the sum of the spin carried by up, down, and strange quarks (Δu, Δd, Δs). Early measurements suggested that quarks contribute only a portion of the nucleon spin, a puzzle that spurred a wide array of theoretical and experimental efforts to quantify the remaining contributions, including the role of gluon polarization and orbital motion. The prevailing view among practitioners is that a substantial share comes from orbital angular momentum and, depending on the energy scale, from gluon spin as well, with the exact balance still a topic of active research. See the framework in the context of Quantum Chromodynamics and the parton model for a deeper foundation: parton distribution functions and their spin-dependent counterparts.

Spin structure in the nucleon

Decomposition of spin

The total spin of a nucleon arises from multiple internal motion channels. Quark spins contribute via the axial-vector charges Δu, Δd, and Δs, while gluons carry spin as well, and both sectors support orbital angular momentum. The contemporary picture emphasizes that L_q and L_g are not merely small corrections but essential parts of the full spin budget, especially when probed at higher momentum transfers. For a theoretical treatment, see the operator product expansion and the sum-rule analyses that connect moments of spin structure functions to fundamental charges and to the angular momentum content of partons: Operator product expansion and Bjorken sum rule.

Structure functions g1 and g2

Spin structure functions g1(x,Q^2) and g2(x,Q^2) are extracted from polarized deep inelastic scattering (DIS) of leptons on nucleon targets. The function g1 reflects the alignment of quark spins with the nucleon spin, and its integral over x at fixed Q^2 is related to the net quark-spin contribution ΔΣ through a sum-rule analysis: the first moment of g1 is connected to axial charges and to the sea of quark spins inside the nucleon. The function g2 contains additional information about quark-gluon correlations and higher-twist effects beyond the simple parton model; it is partly constrained by the Wandzura-Wilczek relation, which expresses part of g2 in terms of g1, though genuine twist-3 contributions signal more complex dynamics. For foundational material, see deep inelastic scattering and the dedicated spin structure function discussions: g1 and g2.

Sum rules and interpretation

Key theoretical results tie the measured structure functions to fundamental properties. The Bjorken sum rule relates the difference between proton and neutron (or equivalently, isospin partners) first moments of g1 to the axial charge (g_A) in weak interactions, offering a stringent test of QCD. The Ellis–Jaffe sum rule provides a benchmark for the quark-model expectation of the spin content under certain assumptions, and deviations from it illuminate the roles of strange quark polarization and gluon contributions. These sum rules, together with their QCD corrections, provide a bridge between experimental data and the spin decomposition of the nucleon: Bjorken sum rule and Ellis–Jaffe sum rule.

Experimental history and key results

The defining moment came with the European Muon Collaboration (EMC) measurements in the late 1980s, which indicated that quark spins contributed a surprisingly small fraction of the proton’s spin. This sparked the so‑called spin puzzle, initiating a broad program of polarized DIS experiments worldwide. Subsequent measurements at facilities such as SLAC, CERN (including the Spin Muon Collaboration, or COMPASS), DESY, and Jefferson Laboratory refined the picture of how quark spins contribute and began to probe gluon polarization through high-energy polarized collisions. The Relativistic Heavy Ion Collider (RHIC) has been instrumental in directly constraining ΔG through polarized proton–proton collisions, helping to delineate the gluon’s role in the spin budget. In parallel, advances in lattice QCD and other nonperturbative approaches have provided ab initio estimates for ΔΣ and the orbital components. See the main experimental programs at RHIC (Relativistic Heavy Ion Collider), COMPASS (experiment), and Jefferson Lab for the major milestones, and the ongoing work toward an electron–ion collider for future precision: Electron–ion collider.

Theoretical framework and ongoing debates

Parton model, QCD, and factorization

The spin structure of the nucleon is analyzed within the parton model extended to QCD. Spin-dependent parton distribution functions describe the probability of finding a parton with a given momentum fraction and spin orientation inside the nucleon. Factorization theorems allow separating short-distance, perturbatively calculable parts from long-distance, nonperturbative PDFs, so that experimental observables can be related to these universal functions. This theoretical backbone is augmented by lattice calculations and continuum methods to estimate moments and to illuminate the interplay between spin and orbital momentum.

Gluon spin and orbital angular momentum

While ΔG is constrained by polarized high-energy collisions, the precise partitioning of spin between gluon spin and orbital motion remains a central question. Evidence suggests that orbital angular momentum, both from quarks and gluons, plays a nontrivial role, particularly when considering the spin sum rules across different Q^2 scales. Generalized parton distributions (GPDs) and transverse-momentum-dependent distributions (TMDs) provide more differential information about how angular momentum is distributed in momentum and position space. See the discussions of lattice QCD and Generalized parton distributions for the modern toolkit to address these questions.

Controversies and interpretation

As with many areas at the interface of experiment and theory, there are debates about how to interpret the data, the role of higher-twist effects in g2, and the extent to which lattice results reflect physical quark-gluon dynamics at accessible scales. Some viewpoints emphasize that the observed spin deficit in ΔΣ reflects a substantial contribution from orbital angular momentum, while others argue for nonperturbative mechanisms or scheme dependencies in extracting ΔΣ from data. The broad consensus remains that QCD provides the correct framework, with continued refinements in both measurements and theory.