The Proton Spin Puzzle

The total spin of the hadrons can be determined trivially by simply adding up the 1/2 spins of its valence quarks, with possible plus and minus values for each one. Each combination of plus or minus 1/2 spins adds up to a total spin, and each possible sum of spins for the valence quarks equals the possible total spins of hadrons with those valence quarks. Minimal values for a set of valence quarks are more stable, so protons and neutrons having a minimal possible combination of spins (i.e. they have spin 1/2 equal to 1/2 + 1/2 -1/2) since it is stable. All non-minimal spin sums are unstable hadrons

Surprisingly, however, this simple formula doesn't reflect the actual spin of the full array of valence quarks, sea quarks, and gluons that add up to spin-1/2 in an actual proton. Reality gets to the same result, but in a much more complicated way.

A new PhD dissertation (250 pages) exhaustively examines this puzzle and uses a novel method to try to solve it with a formula (i.e. analytically) rather than with a numerical approximation, extrapolating down to the 3 color, 6 flavor reality, from more complex models with larger numbers of colors and flavors.

The proton spin puzzle denotes the challenge of describing the proton's spin in terms of the angular momenta of the quarks and gluons which comprise it. These quarks and gluons carry a fraction x of the proton's momentum. Contributions from small-x quarks and gluons, which only possess a little of the proton's momentum, are difficult to measure, since this requires very high energy experiments. Furthermore, early theoretical work in the 1990s predicted substantial contributions to the proton spin from these small-x particles. We need theoretical control over this corner of phase space in order to resolve the spin puzzle.

In this dissertation, we build upon an existing framework for studying spin at small-x. Previously, several sets of small-x evolution equations were derived in this formalism -- one in the large-N(c) limit and one in the large-N(c) & N(f) limit. Here N(c) and N(f) are the numbers of quark colors and flavors [ed. there are three colors, three anti-colors, and six flavors in the Standard Model]. These equations were numerically solved but no analytic solutions had been found. In this dissertation we detail the construction of such analytic solutions, first in the large-N(c) limit and then in the large-N(c) & N(f) limit, after deriving an important correction to the existing large-N(c) & N(f) equations due to the contributions of quark-to-gluon transition operators.

From the solutions constructed here, we can predict the behavior of the quark and gluon helicity distributions at asymptotically small-x (and large-N(c) or large- N(c) & N(f)), both as a general power law and further as explicit analytic expressions in the asymptotic limit. Our solutions also allow us to predict all four polarized DGLAP anomalous dimensions in the same limits, yielding expressions exact to all orders in the strong coupling. The expansions of our predictions agree completely with the full extent of existing finite-order calculations, to three loops.

Jeremy Borden, "Searching for the Proton's Missing Spin: Small-x Helicity Evolution Equations and Their Analytic Solutions" arXiv:2603.20906 (March 21, 2026).

The dissertation's introduction does a good job of laying out the puzzle:

A relatively naive — but in some ways still very successful — model takes the proton to be made of three quarks (the general class of particles we call baryons are described in this way as bound states of three quarks). The quarks in this model are nonrelativistic and, like the proton, are spin-1/2 fermions. In such a model, it is easy to intuitively understand the proton’s spin — that is, its intrinsic angular momentum. Two of the constituent quarks have their spins pointed in the direction aligned with the proton’s spin, while the third constituent quark’s spin is in the opposite direction, as visualized in Fig. 1.1. 

In this model 100% of the proton’s spin is accounted for by the spin of the quarks. Perhaps unsurprisingly, more sophisticated models of the proton were also developed (see e.g. the bag model of [10]). But even these sophisticated models which tried to accommodate more complicated phenomena like special relativity and confinement still predicted that a substantial quantity of the proton’s spin must be carried by the quark spins, typically somewhere on the order of 60%. 

Figure 1.1: Naive quark model of the proton P with two quark q spins aligned and one anti-aligned relative to the proton spin. 

Then in the late 1980s, the European Muon Collaboration (EMC) utilized polarized muon-proton scattering experiments to measure the net amount of the proton’s spin carried by the quark spins. The shocking result was a measured value of around 6% [11,12]. Even allowing for the maximal experimental uncertainties, this was an irreconcilable difference compared to theoretical predictions. Thus began the proton spin puzzle. The majority of the proton’s spin could not be accounted for by the theoretical models of the time. 

The good news, however, is that today we have many powerful tools at our disposal to better understand the rich internal structure of the proton, chief among them Quantum Chromodynamics (QCD). Beginning in the 1970s, QCD began to emerge as the presumptive theoretical description of the strong force — the force that binds protons and neutrons together in atomic nuclei, and as would come to be understood, the force that governs the complicated internal structures of the proton and neutron themselves, along with a host of other strongly-bound particles. 

QCD is a non-abelian SU(Nc) gauge theory which describes the fundamental degrees of freedom of the strong force as quarks and gluons. The quarks of QCD are spin-1/2 fermions with fractional electric charges, although they are not exactly the same as the ‘constituent’ quarks shown in Fig. 1.1. There are six flavors of quarks in the Standard Model, varying in their masses and electric charges. In addition to electric charge, the quarks are also charged under the strong force. This color charge comes in Nc = 3 varieties called red, green, and blue (and the antiparticles of the quarks, the antiquarks, can carry anti-red, anti-green, or anti-blue color charge). The quarks form a color triplet and transform under the fundamental representation of SU(Nc). Meanwhile gluons — the strong-force-carriers — are spin-1 bosons that also carry a net color charge, a combination of color and anti-color (the color octet), and transform under the adjoint representation of SU(Nc). Notably the fact that gluons are charged under the strong force means they can interact with other gluons. This is a crucial difference from abelian theories like quantum electrodynamics [13] where the force-carrying particles (photons) do not self-interact. 

Among the consequences of the gluonic self-interactions in QCD is asymptotic freedom [14,15], a remarkable property that tells us the particles of QCD interact very weakly at short distances (or large momentum transfer). This has critical implications for perturbative QCD calculations. The strong coupling — the physical parameter that controls the strength of the force — becomes relatively small at these short distance scales, and so we have a small dimensionless parameter in which we can make a reliable perturbative expansion. This perturbative regime of QCD is the backdrop for this entire dissertation and so the applicability of perturbation theory is critical here. Note however, that the running of the strong coupling — that is, how the coupling changes with momentum scale — also has important implications in the low-momentum/long distance regime. Whereas at high momentum scales the coupling is smaller and we can employ perturbation theory, at low momentum scales the coupling becomes very strong and perturbative methods break down. 

This also hints at the perplexing notion of confinement [16], whereby free color charges cannot be isolated. They are always confined in color neutral combinations. A complete theoretical understanding of confinement is still lacking. 

A particularly useful model of the the proton (and other hadrons) at high energy is Feynman’s parton model [17], where the proton is taken to be a system of point-like particles called partons. Particularly effective in understanding the results of deep inelastic scattering (DIS) of electrons and protons1 at SLAC [18], the model treats the proton (in a frame where the proton is moving ultrarelativistically) as a collection of these co-moving point-like partons which do not interact with each other. 

When colliding with the electron, the system of partons interacts with the electron probe incoherently. Feynman was agnostic about what particles these partons might be but the interpretation that emerged, and the one we still use today, is that they are the quarks and gluons of QCD. The parton framework serves as a powerful tool for understanding how the properties of the proton emerge from the properties of the intrinsic QCD degrees of freedom. But note that we are not limited to the three quark model like that in Fig. 1.1. Instead we can have many partons and we will often label them with the Bjorken-x variable, which corresponds to the longitudinal momentum fraction of a given parton relative to the parent proton. Intuitively, a parton could have as little as zero longitudinal momentum (x = 0) and as much as the full momentum of the proton (x = 1) and so 0 < x < 1. 

The modern picture of the proton’s structure that has emerged holds that there are indeed three quarks that live at relatively large x (that is, close to x = 1) — these are the valence quarks. But we also have a rich sea of quarks and antiquarks at smaller values of x. These sea quarks can fluctuate in number, as particle-antiparticle pairs are created or annihilated, and the interactions among the sea quarks are mediated by gluons, which can themselves split into more gluons or recombine with each other. The interior structure of the proton is thus much less trivial than the naive diagram in Fig. 1.1, but could instead look (for illustrative purposes only) more like the representation in Fig. 1.2.

Figure 1.2: A more complicated but realistic illustration of the proton’s structure. In addition to the three valence quarks (the large spheres), we now have a sea of quarks and antiquarks (the smaller colorful spheres) along with many gluons (the corkscrew lines). The spins of the particles are not represented here, but each quark and gluon can contribute its spin — and also its orbital angular momentum — to the proton’s spin. 

To describe the spin of the proton, we can make the following decomposition: 

Sq +Lq +SG+LG = 1/2. (1.1) 

This is the Jaffe-Manohar sum rule [19]. Eq. (1.1) says that we can break the spin of the proton, which is 1/2 in units of ℏ, into the spins S and orbital angular momenta (OAM) L of the quarks q and gluons G.