Process-independent observability of the gauge-invariant canonical orbital angular momentum in the nucleon spin decomposition

The spin of the nucleon can be decomposed into quark and gluon spin and orbital angular momentum (OAM) in two physically inequivalent gauge-invariant ways: a kinetic ('mechanical') decomposition built from local operators, and a canonical decomposition generalizing the Jaffe-Manohar sum rule. Wakamatsu (Phys. Rev. D 87, 094035, 2013) proved the gauge- and path-independence of the gluon spin at one loop within general axial gauges (straight Wilson-line paths) and argued against the view that there are infinitely many physical decompositions, while explicitly leaving open whether the generalized canonical OAM of the canonical decomposition — realizable as the $(\boldsymbol{b}\times\boldsymbol{k})$ moment of Wigner distributions and generalized transverse-momentum-dependent parton distributions (GTMDs) with staple-shaped gauge links — can be extracted as a process-independent observable. Later work settled the operator level: staple links give the canonical OAM, straight links the kinetic OAM, and their difference (the potential OAM, a closed Wilson loop interpretable as final-state-interaction torque) is confirmed on the lattice. The source question itself, however, remains open: the literature contains no factorization or universality theorem making the canonical OAM process-independently extractable.

Background

A central question of hadron structure, ever since polarized deep-inelastic scattering revealed that quark spins carry only a small fraction of the nucleon's spin, is how the nucleon's spin-1/2 is built from the spins and orbital angular momenta (OAM, the rotational motion of constituents about the nucleon center) of its quarks and gluons. Two classic answers coexist. The Jaffe-Manohar decomposition (1990) is the parton-model sum rule $\tfrac{1}{2}=\tfrac{1}{2}\Delta\Sigma+\Delta G+L_q^{\rm can}+L_g^{\rm can}$, natural in light-cone quantization (the formulation in which the nucleon's constituents appear as quasi-free partons, i.e. the quarks and gluons seen by hard probes), where $\Delta\Sigma$ and $\Delta G$ are the quark and gluon helicity contributions and $L_{q,g}^{\rm can}$ are canonical OAMs; its operators are gauge dependent as they stand. The Ji decomposition (1997), $\tfrac{1}{2}=\tfrac{1}{2}\Delta\Sigma+\sum_q L_q^{\rm kin}+J_g$, uses local gauge-invariant operators, but its gluon total angular momentum $J_g$ cannot be split into spin and OAM pieces within that framework.

The obstacle is the one familiar from QED textbooks: the photon's spin and OAM operators are gauge dependent, and the same is worse in QCD. Chen et al. (2008) proposed restoring gauge invariance by splitting the gauge potential as $A_\mu=A_\mu^{\rm pure}+A_\mu^{\rm phys}$, where $A^{\rm pure}$ is a pure gauge and $A^{\rm phys}$ the physical (transverse) component; in the non-abelian theory $A^{\rm phys}$ can only be written in closed form with a Wilson line (gauge link) — a path-ordered exponential of the gauge field along a contour — and the choice of contour is a choice of path. Since infinitely many paths connect two spacetime points, a controversy arose over whether this yields infinitely many physically distinct nucleon-spin decompositions, which would make the spin/OAM split arbitrary.

Wakamatsu (2013) addressed this within the general axial gauge, the class of gauges $n^\mu A_\mu=0$ for a constant four-vector $n^\mu$, containing the temporal, light-cone, and spatial axial gauges and equivalent to straight Wilson-line paths running to spacetime infinity. He identified two physically inequivalent gauge-invariant decompositions: (I) the mechanical type (a gauge-invariant extension of Ji's, in which the gluon angular momentum can be split) and (II) the canonical type (a gauge-invariant extension of Jaffe-Manohar). Computing the one-loop anomalous-dimension matrix (the quantities governing the scale evolution) for the quark and gluon longitudinal spins, he found it independent of $n^\mu$, which proves gauge- and path-independence at one loop; the axial gauges' spurious poles were handled with the generalized $n^*$ prescription of Mandelstam, Leibbrandt, and Gaigg et al. But the proof is perturbative and restricted to straight paths, and, decisively, the generalized canonical OAM of decomposition (II) had no established observable status: it connects to Wigner distributions (quantum phase-space quasi-probabilities) and transverse-momentum-dependent parton distributions (TMDs), whose gauge links are process-dependent staple-shaped contours rather than straight lines, and the factorization and universality of such distributions with quantum-loop effects was, and partly remains, under debate. His conclusion states the open question plainly: is process-independent extraction of the canonical OAM possible?

The modern framework in which the question now lives is this: quark and gluon OAMs are the $(\boldsymbol{b}\times\boldsymbol{k})$ moments of Wigner distributions, equivalently generalized TMDs (GTMDs), which depend on both transverse position $\boldsymbol{b}$ and transverse momentum $\boldsymbol{k}$. A staple-shaped gauge link extending to light-cone infinity yields the canonical OAM; a straight link yields the kinetic OAM; their difference, the potential OAM, is a closed Wilson loop that Burkardt interpreted as the torque from final-state interactions, a picture confirmed numerically in lattice QCD. What is not settled is whether the canonical OAM so defined can be extracted from experiments independently of the process used — the question this problem poses.

Problem Statement

The open problem left by Wakamatsu (2013) is:

Is the gauge-invariant generalized canonical orbital angular momentum of quarks and gluons in the nucleon — the orbital piece of the canonical-type (Jaffe-Manohar) decomposition, realized as the $(\boldsymbol{b}\times\boldsymbol{k})$ moment of Wigner distributions and generalized transverse-momentum-dependent distributions (GTMDs) with staple-shaped Wilson lines (path-ordered exponentials of the gauge field along a staple-shaped contour) — a process-independent observable that can be extracted from hard-scattering experiments?

The question may be answered affirmatively or negatively.

Related open aspects that may inform an answer, without being independent requirements, include the nonperturbative status of the gauge- and path-independence properties established perturbatively at one loop for straight Wilson paths in general axial gauges, and the state of concrete candidate hard processes — e.g., double-spin asymmetries in coherent diffractive dijet production or exclusive meson production, as pursued for the Electron-Ion Collider — which have so far been computed only at leading order.

References

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