Heavy-quark spin correlations could probe crowded gluon fields

Two spins, one quantum state

When a high-energy proton collides with another proton or a nucleus, the interaction can produce a quark and its antiquark. Their spins can be entangled: the joint state cannot be described as a statistical mixture of independent spin states. Could those correlations also reveal the behavior of the gluons that mediate the strong force?

In a November 2026 preprint, Michael Fucilla calculates how the surrounding gluon field affects the pair’s spin correlations. The work is theoretical, not a report of newly measured entanglement or gluon saturation. It also proposes ways to infer otherwise inaccessible quark spins from the particles that emerge inside jets.

Calculating nearly opposite-moving pairs

The calculation concerns pairs moving almost back-to-back in the plane perpendicular to the beams. Each quark’s transverse momentum must greatly exceed both the pair’s small momentum imbalance and the target’s saturation scale, which characterizes when dense-gluon effects become important.

Fucilla uses a high-energy description in which the target appears as a thin sheet of gluon fields. The analysis focuses on small-x gluons: gluons carrying a small fraction of the target’s longitudinal momentum. Their distributions include information about transverse momentum and linear polarization, even when the colliding beams have no overall polarization.

From the production amplitudes, the paper constructs the pair’s spin-density matrix—the mathematical description of its spin state. It then evaluates entanglement using concurrence, a quantity ranging from zero for separable states to one for maximal entanglement. Other diagnostics capture different quantum properties; they are not interchangeable measures of entanglement.

A massive-quark angular pattern sensitive to gluon density

The saturation-sensitive prediction concerns massive quarks and the angle between the pair’s relative transverse momentum and its small residual imbalance. Linearly polarized gluons introduce a cosine-of-twice-the-angle contribution to the spin-density matrix. This produces angular variation in concurrence and other quantum-information quantities.

For inclusive production, where the target’s final state is not fixed, the effect depends on a combination of six gluon distributions: three unpolarized distributions and three linearly polarized partners.

In the dilute approximation used for comparison, the relevant polarized-to-unpolarized ratio equals one. The resulting angular dependence is independent of the imbalance magnitude within that approximation. Including gluon saturation changes the ratio; the paper’s nuclear-model examples show substantial suppression of the modulation when the imbalance is comparable to the saturation scale.

This does not mean saturation simply destroys entanglement. Rather, it changes how entanglement varies with the collision geometry. Some configurations remain entangled while others are separable. Moreover, averaging the production density matrix over the relevant angle removes the polarized-gluon contribution at the accuracy studied, concealing this particular sensitivity.

The quark mass is essential to this proposed probe. In the massless limit of the inclusive leading-order calculation, the polarized-gluon angular terms vanish. The remaining gluon-distribution factor cancels from the normalized spin state, which therefore loses its gluon-distribution dependence. Light-quark entanglement can still be investigated, but not through this saturation-sensitive modulation in that limit.

The numerical examples use a model suited to a large nucleus. They illustrate the mechanism rather than provide precision predictions for a particular detector. Reliable proton predictions would require further treatment of how gluon distributions evolve toward smaller x.

Diffraction produces a restricted spin state

The paper separately examines coherent diffraction, in which the target remains intact and is separated from the produced system by a rapidity gap—a region without produced particles.

Within the leading high-energy, near-forward treatment, fixing the target’s final state and averaging over two incoming gluon polarizations restricts the spin-density matrix to rank at most two. Each individual quark is unpolarized. Together, these properties make the pair’s state equivalent, after changing local spin axes, to a mixture of at most two maximally entangled Bell states.

For this special family, entanglement and the ability to violate a Bell–CHSH inequality coincide. That equivalence is not true for arbitrary mixed quantum states. The calculated amplitudes predict both properties across the interior of the considered phase space, excluding the limiting momentum-sharing endpoints. This is a theoretical statement about the spin state, not an experimental Bell test.

Hadronic diffraction has an additional experimental complication. Soft interactions involving proton spectators—the proton constituents outside the hard production process—can destroy the rapidity gap or break up the target. Predictions therefore require an uncertain gap-survival correction. This correction mainly affects the event rate and should cancel from normalized spin correlations only if it is independent of spin. That cancellation cannot be assumed without the condition. Selecting an intact target and reconstructing heavy-quark spin information add further demands.

Reading spins through fragmentation

Quarks lighter than the top quark form hadrons before decaying, complicating spin measurements. Fucilla proposes exploiting fragmentation: the process through which quarks produce observable hadrons.

The Collins effect connects transverse quark spin to hadron directions within a jet. Correlations between hadrons in opposite jets could therefore reveal transverse spin correlations. Additional dihadron observables would be needed to access longitudinal components and complete the massive-quark spin information.

The light-quark proposal is distinct from the massive-quark saturation probe. For massless quarks in the selected leading-order production channel, a particular transverse spin coefficient equals the concurrence. A suitably selected double-Collins asymmetry could therefore act as an entanglement witness. The measured angular asymmetry would also depend on fragmentation functions; it would not directly equal entanglement. Its hadron-angle dependence should not be confused with the saturation-sensitive angle between the pair’s relative momentum and imbalance, whose contribution vanishes in this massless limit.

Charge-selected hadron combinations cancel backgrounds containing final-state gluon jets, but not every competing quark process. Higher-order radiation, competing channels, fragmentation uncertainties and experimental feasibility remain to be quantified. The result supplies two complementary proposals: massive-quark spin correlations as a possible probe of dense-gluon dynamics, and fragmentation-based observables as possible witnesses of light-quark entanglement—not yet demonstrated measurement methods.