The EMC discovery that the spins of the quarks carry only about a third of the proton's spin left the quantitative decomposition of the proton's spin 1/2 among quark spin, gluon spin, and quark/gluon orbital angular momentum (OAM) as a central open problem of hadronic physics. Myhrer and Thomas (Phys. Lett. B 663 (2008) 302) argued that relativistic valence-quark motion, the one-gluon-exchange hyperfine exchange current, and the chiral pion cloud jointly reduce the model quark spin to the observed value, predicting that the missing spin resides mostly in valence-quark OAM with a small gluon polarization. The problem remains open today: global QCD analyses determine a positive gluon helicity contribution only for momentum fractions x above roughly 0.05, with the small-x region unconstrained; the quark and gluon orbital angular momenta have no direct experimental determination; and the pion-cloud probabilities and the non-perturbative matching of the flavor-singlet axial charge remain constrained only at model level, even as lattice QCD now supplies first-principles benchmarks such as a continuum-limit gluon helicity of 0.231(17)(44) at a scale of 10 GeV squared.

Background

The proton is a spin-1/2 bound state of QCD (quantum chromodynamics, the theory of the strong interaction among quarks and gluons), and a basic question of strong-interaction physics is how its total angular momentum is distributed over the spin and orbital motion of its constituents. In deep-inelastic scattering (DIS), the scattering of an electron or muon off the proton at high momentum transfer, the spin alignment of a parton (a quark, antiquark or gluon) with the proton spin is measured by polarized structure functions. The European Muon Collaboration found in 1988 that the flavor-singlet combination of quark helicities, usually denoted ΔΣ\Delta\Sigma (the singlet axial charge, equal to twice the total quark spin contribution), was far below the naive quark-model expectation of about 1: the modern experimental value is around 0.33. This is the proton spin crisis. The shortfall was first attributed to a large gluon polarization, ΔG\Delta G (the net helicity of gluons in the proton), whose contribution to the measured structure function is obscured by the U(1) axial anomaly, a quantum effect that spoils the naive conservation of the singlet axial current; but experiments at CERN, DESY, Jefferson Lab and RHIC have since shown that gluon spin is too small to explain the whole deficit.

Modern discussions use the gauge-invariant decomposition of Ji:

12=12 ΔΣ+Lq+12 ΔG+Lg\frac{1}{2} = \frac{1}{2}\,\Delta\Sigma + L_q + \frac{1}{2}\,\Delta G + L_g

where LqL_q and LgL_g are the quark and gluon orbital angular momenta, evaluated at a specified renormalization scale (the energy scale at which the separation into quark and gluon contributions is defined). The total quark and gluon angular momenta are accessible through generalized parton distributions (GPDs), non-forward matrix elements of quark and gluon operators measured in deeply virtual Compton scattering, in which a virtual photon scatters off the proton and emerges as a real photon, and in exclusive meson production; their second moments satisfy Ji's sum rule above. The gluon polarization ΔG\Delta G is constrained by polarized semi-inclusive DIS, open-charm production, and jet or pi0 production data in global QCD analyses.

Myhrer and Thomas proposed a specific resolution of the crisis within constituent-quark-model physics (models in which the proton is built from three effective, dressed quarks): (i) relativistic motion of confined valence quarks (a Dirac lower component carrying orbital angular momentum) reduces the quark spin by about one third; (ii) the one-gluon-exchange (OGE) hyperfine interaction, best known from the mass splitting between the nucleon and its first excited state, the Delta resonance, induces a two-quark exchange-current correction that converts roughly 0.15 units of quark spin into orbital angular momentum of p-wave antiquarks, dominantly anti-u; and (iii) the pion cloud, the mesonic dressing required by chiral symmetry (the approximate symmetry of QCD acting independently on left- and right-handed quarks) and described for example in the cloudy bag model, multiplies ΔΣ\Delta\Sigma by a factor of about 0.7-0.8, parametrized by the probabilities PNπP_{N\pi} (a bare nucleon plus one pion, about 0.20-0.25) and PΔπP_{\Delta\pi} (a bare Delta resonance plus one pion, about 0.05-0.10) in the proton wave function. Together these effects reduce the model ΔΣ\Delta\Sigma from about 0.65 to 0.35-0.40, or to 0.21-0.32 after a non-perturbative matching factor (0.6-0.8, from the three-loop anomalous dimension of the singlet axial current) that relates a low-scale model value to the reported experimental quantity, in good agreement with observation. The authors concluded that the missing spin must be carried mainly by orbital angular momentum of quarks and antiquarks, making the orbital structure a prime target of the Jefferson Lab 12 GeV program and of the future Electron-Ion Collider (EIC), a planned high-luminosity collider with polarized electron and proton beams designed to reach gluon-dominated kinematics down to momentum fractions of order 10^-4.

What the authors themselves flagged as unresolved: important measurements of the gluon polarization were still underway, although experiments had shown it most likely too small to account for the whole deficit; the non-perturbative matching factor relating the low-scale model value of ΔΣ\Delta\Sigma to the reported experimental quantity lacks a controlled determination, its evaluation through even three-loop perturbation theory being 'at best semi-quantitative'; and the missing spin is expected to reside in the orbital angular momentum of quarks and antiquarks, which remains to be measured and mapped, the authors noting that exploring this angular momentum is a major aim of the Jefferson Lab 12 GeV program. By contrast, they treated the one-gluon-exchange versus pion share of the nucleon-Delta mass splitting as settled: chiral analyses of lattice QCD calculations of the N and Delta masses as a function of quark mass indicate that pion effects contribute 50 MeV or less of the observed 300 MeV mass difference, so that the one-gluon-exchange and pion-cloud corrections can be combined without significant double counting. The pion-cloud probabilities PNπP_{N\pi} and PΔπP_{\Delta\pi} enter as model inputs constrained by data such as the d-bar minus u-bar sea asymmetry. Lattice QCD, the first-principles numerical simulation of QCD on a discrete space-time grid, now supplies benchmarks for several of these quantities but has not closed them.

Problem Statement

The single scientific objective is a complete, quantitatively validated account of how the proton's spin 1/2 is built from its constituents, completing the program whose mechanisms Myhrer and Thomas identified. Specifically, determine, with controlled uncertainties and at a stated factorization scale and scheme (the conventions separating short-distance from long-distance physics), each component of the decomposition

12=12 ΔΣ+Lq+12 ΔG+Lg\frac{1}{2} = \frac{1}{2}\,\Delta\Sigma + L_q + \frac{1}{2}\,\Delta G + L_g

where ΔΣ\Delta\Sigma is the singlet axial charge (twice the quark spin contribution), LqL_q the orbital angular momentum of quarks and antiquarks, ΔG\Delta G the net gluon helicity, and LgL_g the gluon orbital angular momentum, such that: (1) the net gluon helicity and its distribution ΔG(x)\Delta G(x) are determined precisely enough that the gluon contribution to the proton spin is established with quantified uncertainties; (2) the orbital angular momentum carried by quarks, antiquarks and gluons is determined directly, through generalized parton distributions or equivalent observables; and (3) the dynamical explanation is constrained by observables within the stated scheme: the pion-cloud probabilities PNπP_{N\pi} and PΔπP_{\Delta\pi} (constrained for example through the d-bar minus u-bar sea asymmetry and transition form factors), and a controlled non-perturbative determination of the matching factor relating the low-scale model value of the singlet axial charge ΔΣ\Delta\Sigma to the physical quantity, so that the predicted chiral and hyperfine reduction of the quark spin is either quantitatively verified or refuted.

Completion standard: the three conditions above are jointly required, and each is an independently settled or refuted sub-claim; a solution that establishes only some of them is a partial result that must state explicitly which conditions it resolves and which remain open. For each component the solution adopts a definite determination at the stated factorization scale and scheme, reports its uncertainty quantitatively, and identifies the residual gap that the determination does not cover. Condition (3) is resolved by either a quantitative verification or a quantitative refutation of the predicted chiral and hyperfine reduction of the quark spin.

References

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