The Proton Spin Crisis and Hidden-Sector Angular Momentum

The proton has spin \(\tfrac{1}{2}\). For forty years after the quark model was established, it was assumed that this spin came almost entirely from the intrinsic spins of the three valence quarks: two up quarks and one down quark, combining to give \(\tfrac{1}{2}\). The sea quarks and gluons were thought to contribute very little.

The European Muon Collaboration measured the quark spin contribution at CERN by scattering polarized muons off polarized protons and mapping out the spin structure function \(g_1(x)\). Their 1987 result, confirmed and refined many times since, is that quark spins carry only about 25–30% of the proton's total spin. Integrating over the full Bjorken-\(x\) range:

\[ \Delta\Sigma = \Delta u + \Delta d + \Delta s \approx 0.30 \]

where \(\Delta q\) is the contribution of each quark flavor's intrinsic spin to the proton spin. This is the proton spin puzzle, sometimes called the spin crisis.

The angular momentum sum rule says the total spin must be accounted for:

\[ \tfrac{1}{2} = \tfrac{1}{2}\Delta\Sigma + L_q + \Delta G + L_G \]

where \(L_q\) is the orbital angular momentum of quarks, \(\Delta G\) is the gluon helicity, and \(L_G\) is the gluon orbital angular momentum. Three decades of polarized DIS, RHIC spin measurements, and lattice QCD have established that gluon helicity \(\Delta G\) is positive but not large enough by itself to close the gap. The quark orbital contribution \(L_q\) is experimentally difficult to access — it requires generalized parton distributions (GPDs), not forward parton distributions — and remains poorly constrained. The puzzle is not fully resolved.

In IDWT the proton is a colour-singlet composite of quark modes in the \(d=3\) and \(d=4\) sectors (Part 8 of the notes). The \(d=3\) sector (\(S^3\)) hosts the down-type quarks; the \(d=4\) sector (\(\mathbb{CP}^2\)) hosts the up-type quarks. The three valence quarks of the proton (uud) reside in the joint \(d=3\)/\(d=4\) sector space, bound into a colour-singlet configuration by the kernel coupling \(g_{3,4}\).

The proton mode is a Dirac spinor of the wave \(\Psi_\infty\). A Dirac spinor has a large (upper) component and a small (lower) component, and the small component carries one unit of orbital angular momentum relative to the large one — it has the opposite parity. For the confined proton mode this small component lives in the sector coordinates, so the proton's angular momentum budget includes orbital angular momentum in the hidden \(d=3\) and \(d=4\) directions. This is intrinsic to the spinor mode, not an admixture added by hand.

The scalar contact coupling between the sector coordinates does not generate this angular momentum and could not: it is spin-independent, even in each relative coordinate, so it connects the ground only to even-angular-momentum spatial correlations (Part 8 §10 of the notes). It sets the proton's size and confinement energy. The spin content — the magnetic moments, the axial coupling, and the orbital angular momentum in the hidden directions — lives in the Dirac spinor structure of the mode. The proton and neutron magnetic moments take the values

\[ \mu_p \approx 2.793\,\mu_N \qquad \mu_n \approx -1.913\,\mu_N \]

with the sign and the \(\mu_p/\mu_n\) ratio fixed by the colour-singlet projector structure of the uud and udd configurations. The absolute magnitudes follow from the equipartition baseline (isovector \(5\), isoscalar \(1\), giving \(\mu_p = 3\), \(\mu_n = -2\)) with each isospin channel reduced by the derived factor \(m/E = 33/35\) — one power for the isovector, two for the isoscalar: \(\mu_p = 3432/1225 = 2.8016\) (\(+0.31\%\)) and \(\mu_n = -2343/1225 = -1.9127\) (\(-0.02\%\)), with zero fitted parameters. The remaining step is deriving the per-channel powers (Part 8 §10).

The spin crisis asks why quark intrinsic spins carry only \(\approx 30\%\) of the proton spin. The sum rule in IDWT has the same form as QCD, but the orbital contribution \(L_q\) has a specific structural identity: it is the orbital angular momentum carried by the lower (small) component of the proton's Dirac spinor mode in the \(d=3\) and \(d=4\) sector directions.

In the standard QCD account, orbital angular momentum is angular momentum of quarks in three-dimensional space inside the proton. In IDWT, there is that contribution, but there is also angular momentum in the hidden sector directions: the lower component of the Dirac spinor carries one unit of orbital angular momentum, in sector coordinates that a \(d=3\) observer cannot directly resolve. When the proton's spin is measured via deep inelastic scattering, the probe couples through the \(d=2\) electromagnetic structure on the observable coordinates. It measures quark spins and quark orbital angular momentum in the observable directions. It does not couple to the orbital angular momentum in the hidden \(d=3\) and \(d=4\) coordinate directions beyond the observable three.

This is the IDWT-specific content: the spin crisis is a consequence of the proton being a \(d=3\)/\(d=4\) composite in a sector space with more coordinates than a \(d=3\) observer can probe. The missing spin is not missing — it is angular momentum in the additional \(d=3\) and \(d=4\) sector dimensions, present in the mode structure but invisible to the electromagnetic probe.

Concretely: the proton lives in the full \(S^3 \times \mathbb{CP}^2\) sector space, which has \(3 + 4 = 7\) total dimensions. A polarized DIS probe couples to the quark fields via a photon vertex on the three observable coordinates. The orbital contribution from the additional four coordinates — the three \(\mathbb{CP}^2\) directions and the relative \(S^3\) phase — is not picked up. The quark spin sum rule measures \(\Delta\Sigma\); it measures the spin in the observable directions; it misses the orbital piece in the hidden sector directions.

A related observable that is cleanly predicted in IDWT is the nucleon axial coupling \(g_A\), which determines the rate of neutron beta decay and the coupling of the axial current to the nucleon. IDWT gives

\[ g_A = \sqrt{\frac{S(n_s+1,\,3)}{S(n_s,\,3)}} = \sqrt{\frac{35}{20}} = \sqrt{\frac{7}{4}} = 1.3229 \]

where \(n_s = 4\) is the mode number (\(n_s = n_d + n_u = 1+3\)) and \(S(n,d) = \binom{n+d-1}{d}\) is the hockey-stick mode count. The PDG value is \(g_A = 1.2723 \pm 0.0023\), a discrepancy of \(+4.0\%\). The formula gives the ratio of successive \(d=3\) mode counts at the seed level — a purely combinatorial quantity from the sector structure, with nothing tuned to the measurement.

The +4.0% discrepancy is a quench of this leading ratio, and its mechanism is derived. The weak transition operator is a kernel-channel object, so the axial response samples the kernel's two angular channels; the spin-scalar channel carries no spin transfer, and its share of a maximally localized (point-like) nucleon's norm is exactly \(P_L = 2/(2\,S(5,3)) = 1/35\) — an addition-theorem count on the sector sphere, with nothing tuned. This gives \(q = 1 - \tfrac43 P_L = 101/105\) and \(g_A = \sqrt{7/4}\cdot(101/105) = 1.2725\), within the measured uncertainty (\(+0.1\sigma\)). The alternative reading — the relativistic small component of a free Dirac mode — is excluded by the data at \(34\sigma\), and excluded a priori by the framework's channel-projection lemma: every observable samples the kernel's channels only, so a free-Dirac decomposition never enters an observable. The same spin-scalar (\(k=0\)) share is the structural home of the angular momentum carried in the hidden directions (Part 8 §10 of the notes).

The structural picture is in place: the proton is a \(d=3\)/\(d=4\) Dirac-spinor composite, the spin-scalar channel share carries angular momentum in the hidden sector directions, and that provides the structural home for the missing spin fraction. The axial quench is derived — \(g_A = \sqrt{7/4}\cdot(101/105) = 1.2725\) from the channel share \(P_L = 1/35\), an exact addition-theorem count (\(+0.1\sigma\)). The one open premise is the composite's maximal localization: the equal-norm-per-state counting behind \(P_L = 1/35\) is exact for a point-like source, and extending the size program's point-like thesis from elementary modes to the three-quark composite is the remaining step.

To turn the spin-crisis identification into a quantitative test, two things are needed. First, the composite-localization premise above, which underwrites both the axial quench and the magnetic-moment magnitudes (the same derived factor \(m/E = 33/35\) enters both). Second, an IDWT prediction for the quark helicity fraction \(\Delta\Sigma\): if the fraction of the proton spin carried by quark intrinsic spins is set by the channel decomposition of the nucleon's norm, then it is predicted and can be compared to the measured \(\Delta\Sigma \approx 0.30\).

The full account is in Part 8 §10 of the notes: doi:10.5281/zenodo.19767493.