Global $Λ$ polarization in heavy-ion collisions at high baryon density

Using the three-fluid dynamics model, this study calculates global $\Lambda$ polarization in Au+Au collisions at high baryon densities (3–9 GeV), successfully reproducing STAR data at 3 GeV and predicting a broad maximum in polarization around 3–3.9 GeV while analyzing the specific contributions of thermal vorticity, meson fields, thermal shear, and the spin-Hall effect.

The Gist

Imagine two giant, spinning tops made of nuclear matter smashing into each other at nearly the speed of light. This is what happens in a heavy-ion collision inside a particle accelerator.

When these massive "tops" (gold nuclei) crash, they don't just stop; they create a swirling, super-hot soup of particles called the Quark-Gluon Plasma. Because the collision isn't perfectly head-on (it's a glancing blow), this soup starts to spin like a giant whirlpool. This spinning motion is called vorticity.

Now, here's the tricky part: Inside this spinning soup are tiny particles called Lambda ($\Lambda$) hyperons. Just like a spinning top has a specific orientation, these particles have a property called spin. The big question physicists are asking is: Does the giant spin of the soup force these tiny particles to line up in a specific direction?

This paper is a detailed prediction of how much these particles line up (polarize) when the collision happens at high baryon density—which is a fancy way of saying "when the soup is packed very tightly with matter," a condition found at lower collision energies.

Here is the breakdown of the paper's story using everyday analogies:

1. The Main Character: The Spinning Whirlpool

Think of the collision as a blender. When you blend a smoothie, the liquid spins. In this experiment, the "liquid" is nuclear matter.

• The Discovery: Scientists already knew that at very high speeds (high energy), the particles in the blender spin a little bit.
• The Mystery: They suspected that if you slow the blender down (lower energy) but pack it tighter (higher density), the spin might get stronger. But, if you slow it down too much, the spin should stop completely because there's no momentum left.
• The Prediction: The authors predict that the "spin alignment" (polarization) will hit a peak (a maximum) at a specific "Goldilocks" energy level—around 3 to 4 GeV. It's like a hill: the polarization goes up as you slow down, reaches the top of the hill, and then starts to go down.

2. The Four Forces at Play

The authors didn't just guess; they calculated why the particles line up. They looked at four different "pushers" trying to align the particles:

• The Thermal Vorticity (The Big Spin): This is the main driver. It's the sheer force of the swirling soup dragging the particles along. Imagine a leaf caught in a whirlpool; the water forces the leaf to face a certain way.
• The Meson Field (The Invisible Magnet): Imagine the soup is filled with invisible magnetic fields generated by the crowded particles. These fields act like tiny magnets, trying to pull the particles into alignment. The paper finds this effect is strong at the edges of the collision but weak in the center.
• Thermal Shear (The Stretching): Imagine stretching a piece of taffy. The different parts of the taffy move at different speeds, creating a "shear." This stretching motion also nudges the particles to align.
• The Spin-Hall Effect (The Traffic Jam): This is a subtle quantum effect where particles moving through the crowded soup get nudged sideways, similar to how cars might drift in a traffic jam.

The Verdict: The "Big Spin" (Thermal Vorticity) is the boss. The other three effects (Magnetism, Stretching, Traffic Jam) are like tiny assistants. In fact, the "Stretching" and "Traffic Jam" effects mostly cancel each other out, leaving the Big Spin as the main reason the particles line up.

3. The "Feed-Down" Problem

There's a catch. The particles we see aren't just the ones born directly in the collision. Many are "step-children" born from the decay of heavier, unstable particles (resonances).

• The Analogy: Imagine you are trying to count how many people are wearing red hats at a party. But, some people take off their hats and give them to others before you look.
• The Result: The paper accounts for this "hat-swapping" (feed-down). They found that this process reduces the overall alignment by about 20%. It's like a tax on the polarization.

4. The Predictions (The Crystal Ball)

The authors used a sophisticated computer model (called the 3-Fluid Dynamics model) to simulate these collisions.

• The Match: When they simulated the collision at 3 GeV, their results matched the real data collected by the STAR experiment at the RHIC accelerator perfectly. This gives them confidence in their model.
• The Forecast: Since we don't have data yet for energies of 3.2, 3.5, 3.9, and 4.5 GeV, this paper acts as a crystal ball. It predicts that:
  • The alignment will be strongest in a broad range between 3 and 4 GeV.
  • The alignment will be stronger in "semi-central" collisions (glancing blows) than in head-on collisions because glancing blows create more spin.
  • The alignment will be slightly different depending on which part of the "soup" you look at (the center vs. the edges).

5. Why Does This Matter?

Understanding this "spin" is like taking an X-ray of the early universe.

• The Vortex: It tells us how the nuclear matter flows and spins.
• The Density: It helps us understand how matter behaves when it is squeezed incredibly tight (high baryon density), similar to the conditions inside a neutron star.
• The Phase Transition: It might help us figure out if matter changes from a "solid" state to a "fluid" state in a sudden jump (first-order phase transition) or a smooth slide (crossover).

Summary

This paper is a roadmap for future experiments. It says: "We have a model that works perfectly for the 3 GeV data. Based on this, we predict that if you look at the next batch of collisions (3.2 to 4.5 GeV), you will see the particles aligning even more strongly, peaking around 3.5 GeV, before dropping off."

It's a story about spinning tops, invisible magnets, and the search for the perfect "sweet spot" where the universe's most extreme matter reveals its deepest secrets.

Technical Summary

1. Problem Statement

Non-central heavy-ion collisions generate enormous angular momentum ($10^3$–$10^4 \hbar$), leading to collective vortical motion in the produced quark-gluon plasma (QGP) or hadronic matter. Through spin-orbit coupling, this vorticity induces a global polarization of emitted fermions, specifically $\Lambda$ hyperons.

• The Gap: While global polarization ($P_\Lambda$) has been measured by the STAR Collaboration at RHIC ($\sqrt{s_{NN}} \ge 7.7$ GeV) and HADES at lower energies, the behavior of $P_\Lambda$ in the high baryon density region ($3 \le \sqrt{s_{NN}} \le 7.7$ GeV) remains debated.
• The Question: Does $P_\Lambda$ rise monotonically as energy decreases, or does it exhibit a peak (maximum) at a specific energy before dropping to zero at the kinematic threshold ($\sqrt{s_{NN}} = 2m_N$)? Previous models offered conflicting predictions (monotonic rise vs. peak vs. rapid fall).
• Objective: To calculate and predict the global $\Lambda$ polarization in Au+Au collisions within the energy range $3 \le \sqrt{s_{NN}} \le 9$ GeV, with a focus on the STAR Fixed-Target (STAR-FXT) and NICA experimental regimes.

2. Methodology

The study utilizes the Three-Fluid Dynamics (3FD) model combined with a thermodynamic approach to particle polarization.

• Simulation Framework:

  • 3FD Model: Describes the collision as three fluids (two projectile/target remnants and one central fireball). It accounts for baryon stopping and the formation of high baryon density.
  • Equations of State (EoS): Simulations were performed using two EoSs featuring deconfinement transitions: a first-order phase transition (1PT) and a crossover.
  • Freeze-out: An isochronous freeze-out is assumed, defined by the time the average energy density in the central region drops to a critical value.

• Polarization Contributions: The mean spin vector of the $\Lambda$ hyperon is calculated by summing four distinct contributions:

  1. Thermal Vorticity ($\varpi_{\mu\nu}$): The conventional contribution arising from the gradient of the inverse temperature four-vector ($\beta_\mu = u_\mu/T$).
  2. Meson-Field Contribution: Arising from the interaction of the $\Lambda$ magnetic moment with the vector meson field ($V_{\mu\nu}$) generated by the baryon current. This is based on the Relativistic Mean-Field (RMF) model.
  3. Thermal Shear (SIP): A contribution induced by the symmetric derivative of the four-temperature (thermal shear tensor $\xi_{\mu\nu}$).
  4. Spin-Hall Effect (SHE): A contribution induced by the gradient of the chemical potential over temperature ($\partial_\nu (\mu/T)$).

• Calculation Details:

  • Feed-down: Contributions from the decay of higher-lying resonances (e.g., $\Sigma^0 \to \Lambda \gamma$) are included, which reduces the final polarization by $\approx 20\%$.
  • Averaging: The global polarization is obtained by integrating the local spin vector over the freeze-out hypersurface and particle momenta, restricted by the experimental rapidity acceptance (approximated via hydrodynamic rapidity $y_h$).
  • Boost Corrections: The spin vector is transformed from the center-of-mass frame to the $\Lambda$ rest frame, accounting for boost terms.

3. Key Contributions

• Comprehensive Energy Scan: The paper provides a systematic scan of collision energies ($\sqrt{s_{NN}} = 3, 3.2, 3.5, 3.9, 4.5$ GeV) with detailed dependencies on centrality and rapidity.
• Inclusion of New Terms: It is one of the first studies to quantitatively evaluate the Thermal Shear (SIP) and Spin-Hall Effect (SHE) contributions in the low-energy, high-baryon-density regime, alongside the traditional vorticity and meson-field terms.
• Meson-Field Impact: It rigorously analyzes how the meson-field interaction modifies the rapidity distribution, specifically flattening the polarization at forward/backward rapidities.
• Predictions for Future Data: The results serve as specific predictions for the upcoming STAR-FXT program and NICA experiments (BM@N and MPD).

4. Key Results

A. Collision Energy Dependence

• Broad Maximum: The calculations predict a broad maximum in global polarization $P_\Lambda$ occurring at $\sqrt{s_{NN}} \approx 3–3.9$ GeV.
  • The exact position of the peak depends on the centrality and the width of the midrapidity observation window.
  • For $b=8$ fm, the peak is around 3–3.5 GeV.
• Comparison with Data: The results at $\sqrt{s_{NN}} = 3$ GeV reasonably reproduce the existing STAR data. The results at 3.2–4.5 GeV are presented as predictions.
• EoS Sensitivity: The 1PT and crossover EoSs yield very similar results for $\sqrt{s_{NN}} > 3$ GeV. Divergence occurs only below 3 GeV, attributed to numerical fluctuations in derivative calculations rather than physical differences.

B. Rapidity Dependence

• Shape: Unlike the AMPT model (which predicts a decrease from midrapidity to forward rapidities at 3 GeV), the 3FD model predicts an increase in $P_\Lambda$ from midrapidity toward forward/backward rapidities. This is attributed to vorticity rings located at the border between participants and spectators.
• Meson-Field Effect: The meson-field contribution significantly flattens the rapidity distribution, particularly in semi-central collisions ($b=6, 8$ fm) at lower energies (3–3.9 GeV). Without this term, the distribution is strongly peaked at forward/backward rapidities.
• SIP and SHE: These contributions are found to be negligible compared to thermal vorticity. Furthermore, the SIP and SHE terms partially cancel each other out.

C. Centrality Dependence

• Trend: $P_\Lambda$ increases with the impact parameter (decreasing centrality) due to the accumulation of higher angular momentum in the participant matter.
• Agreement: The model agrees well with STAR data for very central (3–10%) and semi-peripheral (30–40%) collisions at 3 GeV but slightly overestimates the data in the 10–30% centrality range.

5. Significance and Conclusion

• Validation of Vorticity: The study confirms that the global polarization is a robust probe of the collective vortical motion in high baryon density matter.
• Role of Meson Fields: The inclusion of the meson-field contribution is crucial for accurately describing the rapidity dependence of polarization, suggesting that vector meson fields play a non-negligible role in spin alignment at low energies.
• Negligible Higher-Order Effects: The finding that SIP and SHE contributions are small and mutually canceling simplifies the theoretical description of polarization in this energy regime, suggesting that thermal vorticity and meson fields are the dominant drivers.
• Experimental Guidance: The prediction of a broad maximum around 3–3.9 GeV provides a clear target for the STAR-FXT and NICA experiments to verify the existence of the peak in global polarization, which would confirm the complex interplay of vorticity and baryon stopping in the QCD phase diagram.

In summary, this paper establishes a robust theoretical baseline for interpreting low-energy heavy-ion collision data, predicting a peak in $\Lambda$ polarization driven by thermal vorticity and modulated by meson fields, while ruling out significant contributions from thermal shear and spin-Hall effects in this regime.