
QUANTUM CHROMODYNAMICS HAS HEADS SPINNING
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 particlelike 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 highmomentum 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 twoweek workshop "Orbital Angular Momentum in QCD" was held at INT on February
517, 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 highmomentum
transfer processes and QCD factorization, dynamical models of nucleon structure, Lattice QCD,
and electromagnetic and protonproton 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 (http://www.int.washington.edu/PROGRAMS/1249w/)
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 highmomentum 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 deepinelastic 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 "gaugeinvariant 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
fastmoving nucleon are projected on particlelike 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 nonperturbative 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 (imaginarytime) correlation functions. The results
show a large difference between u and dquark 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 flavorsinglet
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 particlebased wave function description that emerges in lightfront quantization. It gives
access to quark and gluon angular momentum equivalent to the JaffeManohar operators and
offers a simple mechanical picture of their orbital motion, preserving much of the intuition one
has from the nonrelativistic Schrödinger theory of atoms. Firstprinciple 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 deepinelastic electronnucleon and nucleonnucleon scattering necessarily require
orbital angular momentum in the nucleon's lightfront wave function. A new class of lightfront
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 highmomentum transfer processes were put forward and considered at
the workshop. Inclusive deepinelastic 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 protonproton
data and the COMPASS experiment; a definitive measurement would be possible with a possible
future ElectronIon Collider (EIC). More direct information on quark and gluon orbital angular
momentum could be obtained from deepinelastic processes with identified final states (semiinclusive, exclusive). Certain azimuthal asymmetries in semiinclusive deepinelastic 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 finalstate interaction
effects from angular momentum in the nucleon wave function. Such experiments are performed in
leptonnucleon scattering at HERMES, COMPASS, and JLab and in protonproton 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
http://www.int.washington.edu/PROGRAMS/1249w/.
C. Weiss (weiss@jlab.org), with L. Bland, Z.E. Meziani, G.A. Miller, F. Yuan, M. Vanderhaeghen.
