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Orbital Angular Momentum in QCD
(INT Workshop February 6 - 17, 2012)

Reported by L.C. Bland, Z-E. Meziani, G. Miller, M. Vanderhaeghen, C. Weiss, F. Yuan
  Date posted July 31, 2012


Angular momentum has emerged as a key issue in the quest to understand the internal structure of hadrons on the basis of the fundamental theory of strong interactions, Quantum Chromodynamics (QCD). How does the nucleon's spin arise from the interacting quark and gluon fields or their particle-like quanta? How does the internal motion of quarks and gluons affect basic hadron properties such as form factors, charge and current densities, momentum distributions, and others? Can one probe orbital angular momentum directly in high-momentum transfer scattering processes which resolve the pointlike quark and gluon constituents? The spin densities of quarks and gluons in the nucleon have been studied for more than 20 years, starting with the historic measurements at SLAC and CERN and continued in a series of dedicated experiments (DESY HERMES, CERN COMPASS, Jefferson Lab, BNL RHIC). Quantifying the orbital angular momentum of quarks and gluons has been the focus of the last few years. This has proved to be an extremely challenging problem, stimulating numerous theoretical and experimental investigations and competing proposals. A two-week workshop "Orbital Angular Momentum in QCD" was held at INT on February 5-17, 2012, to review the subject and chart the course for a comprehensive exploration of quark and gluon angular momentum. More than 40 international experts in the areas of high-momentum transfer processes and QCD factorization, dynamical models of nucleon structure, Lattice QCD, and electromagnetic and proton-proton scattering experiments, came together for presentations and intensive discussions. While many individual aspects remain to be investigated, substantial progress was made in identifying current uncertainties and understanding the relation between the different approaches that have been proposed. The emerging consensus is documented in the materials posted on the workshop webpage ( and will be summarized in a future document.

Simply defining the correct quark and gluon angular momentum operators in QCD is a challenging problem that poses many interesting questions. Gauge symmetry necessitates a choice in identifying the physical degrees of freedom, resulting in different equivalent forms of the momentum and angular momentum operators. Several definitions of the quark and gluon angular momentum operators have been proposed, motivated by different ideas about their most natural form. It was realized during the workshop that these definitions should be compared with regard to their practical usefulness: Do the operators have a simple partonic interpretation (see Fig. 1a)? Can they be measured in high-momentum transfer processes or related to other known QCD operators? Can their matrix elements be computed in Lattice QCD?

Taking this point of view, the workshop put the different definitions of quark and gluon angular momentum operators into perspective. The operators proposed by Jaffe and Manohar have a direct partonic interpretation but do not provide a way of measuring the quark and gluon orbital angular momentum or calculating them on the lattice. The local operators of the quark and gluon total angular momentum proposed by Ji can be related to the moment of the nucleon's generalized parton distributions (GPDs) and thus in principle be measured independently. Moreover, their matrix elements can be calculated on the lattice. But these operators do not offer a simple partonic interpretation with a mechanical picture of orbital motion. Another set of operators was obtained by Chen, Goldman, and Wang by imposing the canonical commutation relations as a criterion for the physical degrees of freedom of the gauge field. This choice appears natural for discussing atoms and classical radiation fields in electrodynamics, but its usefulness for hadron structure in QCD remains unclear, as the operators are not directly related to deep-inelastic processes. More general definitions, interpolating between these choices, were proposed by Wakamatsu and by Hatta. An important breakthrough at the workshop was to realize that the various definitions can be related by a technique of "gauge-invariant extension," by which a definition that appears naturally in a particular gauge can be reformulated in an arbitrary gauge. Thus it is now possible to accurately delineate the range of possible choices and the relations between them.

Figure 1: (a) Partonic picture of angular momentum in QCD. The quark and gluon fields in the fast-moving nucleon are projected on particle-like quanta, which are resolved in high momentum- transfer processes. (b) Summary of Lattice QCD results for the u and d quark total angular momentum according to the Ji definition, as a function of the pion mass (plot by S. Syritsyn).

The workshop also considered the evaluation of the quark and gluon angular momentum operators in non-perturbative approaches to nucleon structure in QCD and the role of quark and gluon orbital motion for nucleon observables. Lattice QCD calculations extract the matrix elements of Ji's local operators from Euclidean (imaginary-time) correlation functions. The results show a large difference between u- and d-quark total angular momentum (see Fig. 1b), as found also in models based on chiral degrees of freedom. While considerable challenges remain before a fully realistic description of nucleon structure can be achieved, especially in the flavor-singlet and gluonic sectors, steady progress is being made, and first lattice calculations of gluonic spin structure were reported at the meeting. A complementary approach to nucleon structure in QCD is the particle-based wave function description that emerges in light-front quantization. It gives access to quark and gluon angular momentum equivalent to the Jaffe-Manohar operators and offers a simple mechanical picture of their orbital motion, preserving much of the intuition one has from the non-relativistic Schrödinger theory of atoms. First-principle calculations of hadron structure in this Hamiltonian approach to QCD remain very difficult, but it provides a basis for building phenomenological models which incorporate a large body of experimental information. In particular, this approach shows that the nucleon's Pauli form factor and the observed azimuthal asymmetries in deep-inelastic electron-nucleon and nucleon-nucleon scattering necessarily require orbital angular momentum in the nucleon's light-front wave function. A new class of light-front models was inspired by the idea of "holography" discovered in gauge/string duality and attempts to describe hadron angular momentum structure and spectroscopy in a unified framework.

The most significant remaining challenge is learning how to extract quantitative information on quark and gluon orbital angular momenta from experimental data. Several suggestions for probing angular momentum in high-momentum transfer processes were put forward and considered at the workshop. Inclusive deep-inelastic scattering with polarized nucleons measures the spin- dependent parton densities, which determine part of the nucleon's spin decomposition. While the quark contribution to the nucleon spin is relatively well known by now, the gluon contribution is still subject to considerable uncertainty. New constraints are coming from the RHIC proton-proton data and the COMPASS experiment; a definitive measurement would be possible with a possible future Electron-Ion Collider (EIC). More direct information on quark and gluon orbital angular momentum could be obtained from deep-inelastic processes with identified final states (semi-inclusive, exclusive). Certain azimuthal asymmetries in semi-inclusive deep-inelastic scattering are directly proportional to the quark or gluon orbital angular momentum; the challenge lies in ensuring that a perturbative QCD description is applicable and separating final-state interaction effects from angular momentum in the nucleon wave function. Such experiments are performed in lepton-nucleon scattering at HERMES, COMPASS, and JLab and in proton-proton collisions at RHIC, with both longitudinally and transversely polarized nucleons; major programs are planned with the JLab 12 GeV Upgrade and a future EIC. Exclusive scattering could in principle constrain the quark and gluon angular momentum through its connection with the nucleon GPDs. Here the difficulty lies in the fact that such reactions sample the GPDs in a kinematic region that is far from the domain required to evaluate angular momentum through the Ji sum rule. While exclusive processes provide much interesting information on the nucleon's spatial structure and parton correlations, extraction of the quark and gluon angular momentum will likely be possible only with considerable input from dynamical models. An extensive program of nucleon GPD studies is planned with the JLab 12 GeV Upgrade and a future EIC.

In sum, the INT workshop showed that angular momentum will remain a central issue of QCD and nucleon structure physics of the next decade. Major new impulses will come from the JLab 12 GeV Upgrade and a possible future EIC, as well as from expanding theoretical studies. A complete understanding of the nucleon's angular momentum puzzle will likely require a comprehensive program that combines results of different scattering experiments, phenomenological models, and Lattice QCD calculations. The INT workshop has stimulated intense discussions among the participants and in the wider scientific community and will hopefully enable a more coherent approach to the fascinating problem of angular momentum in QCD. Its impact is demonstrated in several recent publications which were directly or indirectly inspired by the discussions at the meeting, see e.g. X. Ji, X. Xiong, F. Yuan, arXiv:1202.2843, arXiv:1207.5221; X. Chen, arXiv:1203.1288; K.-F. Liu et al., arXiv:1203.6388; M. Wakamatsu, arXiv:1204.2860; X. Ji, Y. Xu, Y. Zhao, arXiv:1205.0156; M. Burkardt, arXiv:1205.2916; H.-C. Kim, P. Schweitzer, U. Yakhshiev, arXiv:1205.5228; C. Lorce, arXiv:1205.6483; Y. Hatta, S. Yoshida, arXiv:1207.5332. The individual presentations and further materials are available on the workshop webpage at

C. Weiss (, with L. Bland, Z.E. Meziani, G.A. Miller, F. Yuan, M. Vanderhaeghen.