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, however, 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).

A Generalized MOND Paradigm For Weak Gravitational Fields And Experimental Confirmation Of GR In Strong Gravitational Fields

This new (published) paper is a provocative generalization of the MOND paradigm, although still not a true "FundaMOND" in the sense of providing a rigorous, fundamental physical explanation for why the phenomenological MOND paradigm is observed in Nature. 

Notably, MOND addresses gravity in the extremely weak gravitational field regime.

The primary author, Robin Eappen, who is a doctoral student in astrophysics, is not well know. But Pavel Kroupa is one of the leading MOND proponents (and is perhaps a little less mainstream in his scientific positions and his manner of advocating for them, than either Mordehai Milgrom who devised MOND in 1983, his now deceased colleague in astrophysics, Jacob Bekenstein, or Stacey McGaugh, one of MOND's leading proponents in the current generation of astrophysicists).

Mass discrepancies in galaxies are empirically known to appear only below a characteristic acceleration scale a(0). 

Here we show that this behaviour is not limited to galaxies: it extends continuously across the full hierarchy of self-gravitating stellar systems, from gas-rich dwarfs and spirals to massive early-type galaxies, and further down to compact stellar clusters. 

We introduce the Milgromian dynamics (MOND) depth index DM, together with dynamical maturity index T = t(cross)/t(H), dynamical collisionality index T(1) = t(cross)/t(relax), with t(cross) being the crossing time, t(H) the Hubble time and t(relax) the median two-body relaxation time, and the MOND acceleration index A = a(bar)/a(0). 

We uncover a well-defined two-dimensional dividing surface in dynamical space. The "dark matter phenomenon" is found only in systems that are both in the deep-MOND regime (a(bar) < a(0)) and collisionless (t(relax) > t(H)), while high-acceleration, collisional systems (a(bar) > a(0), t(relax) << t(H)), including globular clusters and UCDs, show no evidence for a mass discrepancy. 

This clean dynamical separation defines a new, physically motivated classification scheme for stellar systems, unifying galaxies and clusters under one framework. The observed division emerges naturally within the MOND framework and provides a useful diagnostic for examining how different gravitational paradigms account for the origin of the mass discrepancy.

Robin Eappen, Pavel Kroupa, "The MOND Depth Index and Dynamical Maturity Clock: Toward a Universal Classification of Galaxies and Star Clusters" arXiv:2603.18135 (March 18, 2026) (published in 14(2) Galaxies 22 (2026)).

* * *

In other gravity news, a series of three papers (one, two and three) look at gravitational wave observations of extremely strong gravitational fields created by black holes and/or neutron star binary systems to test General Relativity (GR) in this regime in various ways. 

All of this evidence is consistent with GR and more tightly constrains deviations from GR in this strong field context than prior tests of GR.