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 are set by the Dirac small-component structure, a computation that is still open (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 \(CP^2\) 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 composite 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 relativistic quench of this leading ratio. Because the proton mode is a Dirac spinor, its axial matrix element is reduced by the small component: \(q = \langle g^2 - \tfrac13 f^2\rangle / \langle g^2 + f^2\rangle = 1 - \tfrac43 P_L\), where \(P_L\) is the small-component probability and the \(\tfrac13\) is the angular average of the opposite-parity small component. Taking \(P_L = 1/S(5,3) = 1/35\) — one small-component unit relative to the 35 states of the destination level — gives \(q = 101/105\) and \(g_A = \sqrt{7/4}\cdot(101/105) = 1.2725\), within the measured uncertainty. This is the same Dirac small-component structure that carries the missing orbital angular momentum in the hidden directions. The count identity \(P_L = 1/S(5,3)\) reproduces the quench; deriving it from the first-order sector Dirac operator is an open item (Part 8 §10 of the notes).

The structural picture is in place (🔶): the proton is a \(d=3\)/\(d=4\) Dirac-spinor composite, the lower component of the mode carries orbital angular momentum in the hidden sector directions, and that provides the structural home for the missing spin fraction. The axial coupling gives the first quantitative result — \(g_A = \sqrt{7/4}\cdot(101/105) = 1.2725\) from the small-component fraction \(P_L = 1/S(5,3)\), within the measured value. What is not yet in place is the derivation of \(P_L\) — equivalently the small/large ratio of the \(d=3\) nucleon mode — from the first-order sector Dirac operator. The count identity \(P_L = 1/S(5,3)\) reproduces the quench but is not yet derived.

To turn the spin-crisis identification into a quantitative test, two things are needed. First, the derivation of the small-component fraction from the sector Dirac operator, which would fix the magnetic-moment magnitudes and the axial quench together rather than through the count identity alone. 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 how much of the Dirac mode is in the large versus the small component, then it is predicted, not fitted, 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.