Is nucleon spin thermalized in intermediate-energy heavy-ion collisions?

This study demonstrates that in intermediate-energy heavy-ion collisions, spin-thermalized approaches significantly overestimate both global and local nucleon spin polarizations compared to results derived from spin-orbit mean-field potentials in transport simulations.

The Gist

The Big Question: Do Spins "Relax" in a Nuclear Crash?

Imagine two giant, spinning tops (atomic nuclei) smashing into each other at incredibly high speeds. This is what happens in heavy-ion collisions. When they crash, they create a super-hot, messy soup of particles called nuclear matter.

Scientists have been studying this soup for a long time. In the most extreme, high-speed crashes (like those at the Large Hadron Collider), they found that the tiny internal "spins" of the particles (like little bar magnets) seem to get perfectly aligned with the swirling motion of the soup. It's as if the particles are so hot and chaotic that they instantly "relax" and spin in the direction the fluid is swirling. This idea is called the Spin-Thermalized Assumption. Think of it like a crowd of people in a mosh pit; if the pit spins, everyone eventually spins with it.

But here is the problem: This "relaxation" idea works great for high-speed crashes, but when scientists look at slower crashes (intermediate energies), the math starts to break. The "relaxation" model predicts way too much spinning alignment compared to what we might expect.

The New Study: A Different Kind of Simulation

Author Jun Xu decided to test this idea using a different approach. Instead of assuming the spins instantly "relax" to match the swirl, he used a Transport Model (let's call it the "Traffic Simulator").

• The Spin-Thermalized Approach: Imagine a dance floor where the DJ spins the room. Everyone on the floor instantly turns to face the new direction, no matter how fast they were moving before.
• The Transport Model (SIBUU): Imagine a busy highway. Cars (nucleons) are driving, and there is a strong wind (the Spin-Orbit Potential) blowing on them. The cars don't instantly turn to face the wind; they have to physically fight the wind, drift, and change direction over time based on their speed and position.

The Experiment: The "Au+Au" Crash

The author simulated a collision between two Gold (Au) nuclei at a moderate speed (100 AMeV). He tracked every single particle, watching how their coordinates, momentum, and spins changed over time.

He then compared two things:

1. The "Traffic Simulator" (Real Physics): How the spins actually behaved under the influence of the "wind" (spin-orbit force).
2. The "Relaxation Model" (The Assumption): What the spins should be if they instantly aligned with the swirl of the soup.

The Results: The "Relaxation" Model is Overconfident

The findings were quite surprising:

1. The Overestimation: The "Relaxation Model" predicted that the particles would be spinning in alignment with the swirl about 2 to 3 times more strongly than the "Traffic Simulator" actually showed.

   • Analogy: Imagine a weather forecast that predicts a hurricane with 150 mph winds, but when you step outside, it's only a 50 mph gale. The "Relaxation Model" is the overconfident weatherman.

2. The "Wind" Matters More Than the "Swirl": In these slower collisions, the particles aren't just passively drifting in a hot fluid. They are being actively pushed and pulled by a specific force called the Spin-Orbit Potential. It's like a magnetic wind that pushes particles with "spin up" to one side and "spin down" to the other. The simple "Relaxation Model" ignores this wind and assumes the heat does all the work.

3. No "Sign" Problem: In high-speed physics, there's a famous puzzle where the direction of the spin doesn't match the direction of the swirl (the "Sign Problem"). In these slower collisions, the author found that the "Relaxation Model" gets the direction wrong and the magnitude wrong. The "Traffic Simulator" showed a more complex, realistic pattern that the simple model missed.

Why Does This Matter?

• For High-Speed Physics: We thought the "Relaxation" idea was a universal law. This paper says, "Not so fast." It works for the hottest, fastest collisions, but it fails when the collision is a bit cooler and slower.
• For Future Experiments: Scientists are planning to measure the spin of protons in these collisions (using Carbon-12 as a detector). This paper gives them a more accurate prediction: Don't expect the spins to be as perfectly aligned as the old models said. They will be weaker and more complex.

The Bottom Line

In the chaotic dance of a nuclear collision, the particles don't just instantly "go with the flow" of the swirling soup. Instead, they are like dancers being pushed by a strong, invisible wind (the spin-orbit force). If you assume they just relax and spin with the room, you will overestimate how much they are spinning.

In short: The "Spin-Thermalized" assumption is a useful shortcut for high-speed crashes, but for intermediate-speed crashes, it's like using a map of a highway to navigate a winding mountain road—it leads you in the right general direction, but you'll end up far off the path if you don't account for the turns (the spin-orbit forces).

Technical Summary

1. Problem Statement

The study addresses a critical gap in understanding spin dynamics in heavy-ion collisions. While the spin-thermalized assumption (where particle spins are assumed to be completely thermalized in the local vorticity field) has successfully explained hyperon ($\Lambda, \Omega, \Xi$) spin polarizations in ultra-relativistic heavy-ion collisions (dominated by quark-gluon plasma), its validity is challenged at lower (intermediate) collision energies (dominated by hadronic/nucleonic degrees of freedom).

Key issues motivating this study include:

• Experimental observations at lower energies (e.g., by HADES and STAR fixed-target) show that $\Lambda$ spin polarization does not increase monotonically with decreasing energy, contradicting simple thermalization models.
• Theoretical challenges, such as the "sign problem" in longitudinal spin polarization, suggest that non-equilibrium dynamics and specific transport mechanisms may be more relevant than thermal equilibrium assumptions in these regimes.
• There is a lack of quantitative comparison between non-equilibrium transport simulations and thermalized approaches specifically for nucleon spin in the intermediate-energy range (hundreds of AMeV).

2. Methodology

The author employs a comparative approach using two distinct frameworks:

A. Non-Relativistic Spin-Dependent Transport Model (SIBUU)

• Model: A non-relativistic spin- and isospin-dependent Boltzmann-Uehling-Uhlenbeck (SIBUU) transport model.
• Core Equation: The spin-dependent BUU equation for spin-1/2 particles, where the phase-space distribution $\hat{f}$ and single-particle energy $\hat{\epsilon}$ are $2 \times 2$ matrices containing spin-averaged and spin-dependent parts.
• Spin Dynamics: Nucleon spin evolution is driven explicitly by the spin-orbit mean-field potential derived from a Skyrme-type interaction. The equations of motion include a spin precession term ($d\vec{\sigma}_i/dt \propto \vec{h} \times \vec{\sigma}_i$).
• Simulation Details:
  • Collisions simulated: Au+Au at beam energies of 50, 100, and 150 AMeV with impact parameter $b=8$ fm.
  • Spin-orbit coupling coefficient ($W_0$) varied between 80 and 150 MeVfm$^5$.
  • Includes Pauli blocking and energy-dependent differential cross-sections.

B. Spin-Thermalized Approaches

• Assumption: Particle spin is fully thermalized in the local vorticity field.
• Calculations: Spin polarization is calculated post-hoc using bulk properties (density, temperature, velocity) extracted from the SIBUU simulation, but assuming thermal equilibrium.
• Formulas Used:
  • Kinematic Vorticity: $\vec{P} = \vec{\omega}/2T$ (Non-relativistic and Covariant forms).
  • Thermal Vorticity: $\vec{P} = \vec{\varpi}/2$.
  • Spin Vector: $\vec{P} = 2\vec{S}^*$ (including thermal shear effects).
  • Pauli Blocking: Some thermalized calculations include a reduction factor $(1-n_\tau)$ to account for occupied states.

Comparison Strategy:
The study compares the global (integrated) and local (spatial and rapidity-dependent) spin polarizations generated by the SIBUU transport dynamics against those predicted by the thermalized formulas using the same underlying hydrodynamic fields.

3. Key Contributions

• Development of SIBUU: Application of a specific non-relativistic transport model that explicitly tracks nucleon spin degrees of freedom in the intermediate-energy regime, where covariant models are currently lacking.
• Systematic Benchmarking: The first direct comparison of nucleon spin polarization between a non-equilibrium transport model (driven by spin-orbit forces) and thermalized assumptions in intermediate-energy collisions.
• Quantification of Relativistic/Gradient Effects: Demonstration that in this energy regime, relativistic effects and temperature gradients are negligible for spin polarization, isolating the spin-orbit interaction as the dominant mechanism.

4. Key Results

A. Global and Local Polarization Magnitudes

• Overestimation by Thermal Models: All spin-thermalized approaches significantly overestimate the spin polarization compared to the SIBUU transport results.
  • SIBUU: Predicts global polarization ($P_y$) of roughly 2–8% (depending on $W_0$ and energy).
  • Thermalized: Predicts polarization as high as 10–20% in the participant region.
• Spectator Matter: SIBUU predicts negative polarization in spectator matter (due to spin-orbit repulsion), whereas thermalized approaches predict only positive polarization, failing to capture this spatial asymmetry.

B. Mechanism of Generation

• Transport (SIBUU): Polarization is generated dynamically by the spin-orbit mean-field potential. The first term in the potential ($\nabla \times \vec{j}$) dominates, attracting spins aligned with the vorticity to the center and repelling anti-aligned spins to the spectators.
• Thermalized: Assumes equilibrium with the vorticity field. The study finds that temperature gradients and relativistic corrections (covariant terms) largely cancel out in this energy regime, meaning the discrepancy is not due to missing relativistic physics but rather the invalidity of the thermalization assumption itself.

C. Energy Dependence

• SIBUU: Predicts a maximum polarization around 50–100 AMeV. At higher energies, polarization decreases due to stronger spin precession (faster spin rotation) which averages out the alignment.
• Thermalized: Predicts polarization increases with collision energy due to stronger vorticity and weaker Pauli blocking. This contradicts the non-monotonic behavior suggested by SIBUU and hints at the complexity of low-energy data.

D. Longitudinal Polarization ($P_z$)

• Thermalized models predict longitudinal polarization magnitudes ($\sim \pm 10\%$) roughly double those of SIBUU ($\sim \pm 4\%$).
• Both approaches show similar azimuthal angle dependencies, but the thermalized models fail to reproduce the specific spatial structures and magnitudes generated by the non-equilibrium spin-orbit dynamics.

5. Significance

• Theoretical Implication: The study provides strong evidence that the spin-thermalized assumption is invalid for nucleons in intermediate-energy heavy-ion collisions. The spin dynamics in this regime are governed by non-equilibrium transport and spin-orbit coupling rather than local thermodynamic equilibrium with vorticity.
• Experimental Guidance: Since direct nucleon spin polarization data is currently scarce, the paper proposes using protons with known analyzing power (e.g., in $^{12}$C targets) as effective detectors to test these predictions.
• Future Directions: It highlights the need for covariant spin-dependent transport models to bridge the gap between intermediate and ultra-relativistic energies, ensuring a consistent theoretical framework for spin physics across the entire QCD phase diagram.

In conclusion, the paper demonstrates that nucleon spins are not thermalized in intermediate-energy heavy-ion collisions; instead, their polarization is a non-equilibrium phenomenon driven by spin-orbit mean-field potentials, which standard thermalized models fail to capture accurately.