Global polarization of $\Lambda$ hyperons and its sensitivity to equations of state in low-energy heavy-ion collisions

Using the SMASH transport model, this study demonstrates that the hadron resonance gas equation of state successfully reproduces experimental global $\Lambda$ hyperon polarization data at low collision energies, revealing a potential polarization peak around $\sqrt{s_{NN}} = 2.4$ GeV and confirming that helicity polarization vanishes due to space-reversal symmetry.

The Gist

The Big Picture: Spinning Particles in a Cosmic Smackdown

Imagine two heavy nuclei (like gold atoms) smashing into each other at nearly the speed of light. This isn't just a simple crash; it's a chaotic, high-energy "cosmic car crash" that creates a super-hot, super-dense soup of particles called quark-gluon plasma (QGP).

When these collisions happen slightly off-center (like two cars glancing off each other rather than hitting head-on), the debris doesn't just fly apart; it starts to spin. Just like a figure skater pulling their arms in to spin faster, this spinning matter creates a "vortex" or a whirlpool effect.

The big question scientists are asking is: Does this spinning motion make the tiny particles inside the soup (specifically $\Lambda$ hyperons) spin in a specific direction?

The answer is yes. These particles are "globally polarized," meaning they all line up their spins in the same direction, like a crowd of people all turning their heads to look at the same thing. This paper investigates why this happens and what it tells us about the "rules of the road" (the physics) governing this cosmic soup.

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The Mystery: The "Equation of State" (The Recipe)

In physics, the Equation of State (EOS) is like a recipe book. It tells us how the "soup" behaves under different conditions. Is it thick like honey? Thin like water? Does it act like a gas, a liquid, or a weird mix of both?

The authors wanted to know: Does the "recipe" of the soup change how much the particles spin?

To find out, they used a supercomputer simulation called SMASH. Think of SMASH as a massive video game engine that simulates billions of particle collisions. They ran the simulation three times, each time using a different "recipe" (EOS) for the soup:

1. HotQCD: A recipe that assumes the soup is a mix of exotic, high-energy particles (like a super-hot gas).
2. NEOS-BQS: A recipe that tries to account for the fact that there are lots of heavy particles (baryons) in the mix.
3. HRG (Hadron Resonance Gas): A recipe that treats the soup as a gas of standard particles (like protons and neutrons) and their excited cousins.

The Discovery: Only One Recipe Works

Here is the plot twist:

• At high energies: All three recipes gave similar results that matched real-world experiments. It was like three different chefs making a cake that all tasted the same.
• At low energies: The results diverged wildly.
  • The HotQCD and NEOS-BQS recipes predicted that the particles would spin less than what experiments actually saw.
  • The HRG recipe (the "standard particle gas" recipe) was the only one that perfectly matched the experimental data.

The Analogy: Imagine you are trying to predict how fast a spinning top will spin.

• Recipe A says, "It's spinning in a vacuum," and predicts a slow spin.
• Recipe B says, "It's spinning in thick syrup," and predicts a medium spin.
• Recipe C says, "It's spinning in a specific type of oil," and predicts a fast spin.
• Reality check: When you actually spin the top, it spins fast. Therefore, the "oil" (HRG) is the correct description of the physics at low energies.

The "Sweet Spot" (The Peak)

The paper also predicts a "sweet spot" in the energy levels.

• If the collision energy is too low, there isn't enough spin to begin with.
• If the energy is too high, the spin gets diluted.
• The authors found that the maximum spinning (polarization) likely happens at a specific energy level around 2.4 GeV (when using the correct HRG recipe).

Think of this like tuning a radio. You turn the dial (energy), and at a specific frequency, the signal (polarization) is crystal clear and loud. Before this paper, scientists weren't sure exactly where that "station" was. This study suggests it's lower than previously thought.

The "Helicity" Mystery: Why Some Spins Cancel Out

The paper also looked at a more complex type of spin called helicity polarization.

• The Analogy: Imagine a crowd of people running in a circle. If you look at them from the side, they are all running forward. But if you look at them from the top, some are running clockwise and some counter-clockwise.
• The authors found that because the universe has a symmetry (it looks the same if you flip it like a mirror), the "clockwise" and "counter-clockwise" spins cancel each other out perfectly when you average them all together.
• The Result: The net helicity polarization is zero. If experiments ever see a non-zero result here, it would mean there is some other mysterious force at play, not just the spinning of the soup.

Why Does This Matter?

1. It solves a puzzle: For a long time, scientists were confused because their high-energy theories didn't work at low energies. This paper shows that at low energies, the matter behaves like a gas of standard particles (HRG), not a high-energy plasma.
2. It validates the "Vortex" theory: It confirms that the spinning of the particles is indeed caused by the "thermal vorticity" (the heat-induced spin) of the collision, even at energies where we barely produce new particles.
3. It sets a new target: It tells experimentalists exactly where to look (around 2.4 GeV) to find the strongest spinning effects.

In a Nutshell

This paper is like a detective story. The "crime" is the unexpected spinning of particles in low-energy collisions. The "suspects" were three different theories about how matter behaves. By running a high-tech simulation (SMASH), the authors proved that only the "Hadron Resonance Gas" theory (the one treating matter as a gas of standard particles) is the guilty party that explains the evidence. They also found the exact "time of the crime" (the energy level) where the spinning is most intense.

Technical Summary

1. Problem Statement

The global polarization of $\Lambda$ hyperons in non-central relativistic heavy-ion collisions is a well-established phenomenon linked to the thermal vorticity of the quark-gluon plasma (QGP) and the initial orbital angular momentum of the collision system. While this mechanism is well-understood at medium and high collision energies ($\sqrt{s_{NN}} \gtrsim 7.7$ GeV), its behavior at low collision energies ($\sqrt{s_{NN}} < 4.5$ GeV) remains controversial.

Key unresolved issues include:

• Mechanism Validity: Is thermal vorticity still the dominant driver of polarization when collision energies drop below the $\Lambda$ production threshold in nucleon-nucleon collisions ($\approx 2m_N$)?
• Equation of State (EOS) Sensitivity: Experimental data from STAR and HADES show an increasing global polarization as energy decreases, contradicting simple kinetic theory predictions which suggest polarization should vanish. It is unclear how the effective degrees of freedom and the Equation of State (EOS) of the dense matter influence this trend.
• Theoretical Gap: Previous transport model studies have yielded large uncertainties, and the specific dependence of low-energy polarization on the EOS has not been systematically explored within a transport framework.

2. Methodology

The authors employ the SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) transport model to simulate Au+Au collisions at low energies ($\sqrt{s_{NN}} = 2.4, 3.0, 3.5, 3.9, 4.5, 7.7$ GeV).

• Theoretical Framework:

  • Spin Polarization: Calculated using the modified Cooper-Frye formula, which links the spin polarization vector $S^\mu$ to the thermal vorticity $\varpi_{\alpha\beta}$:
    $$S^\mu(x, p) = \frac{1}{8m_\Lambda}(1-f)\epsilon^{\mu\nu\alpha\beta}p_\nu\varpi_{\alpha\beta}$$
    where $\varpi_{\alpha\beta} = \frac{1}{2}[\partial_\alpha(u_\beta/T) - \partial_\beta(u_\alpha/T)]$.
  • Assumptions: The study assumes the system reaches local equilibrium at freeze-out. Non-equilibrium corrections (e.g., second-order gradient terms) are neglected based on high-energy limits where they were found to vanish for global polarization.
  • Mean-Field Potentials: For $\sqrt{s_{NN}} \leq 3$ GeV, finite mean-field potentials (Skyrme and symmetry terms) are included to account for nuclear interactions.

• Equation of State (EOS) Scenarios:
  The authors extract local temperature ($T$) and fluid velocity ($u^\mu$) from the energy-momentum tensor and baryon current, applying three distinct EOS to test sensitivity:

  1. HotQCD: A crossover EOS valid for zero net baryon density (baryon-free).
  2. NEOS-BQS: A crossover EOS extended to finite baryon density via Taylor expansion.
  3. HRG (Hadron Resonance Gas): A purely hadronic EOS, appropriate for the low-temperature, high-baryon-density regime.

• Numerical Setup:

  • Simulations cover 20–50% centrality (impact parameters $b \in [6, 9]$ fm).
  • Event-by-event averaging over $>10^5$ events per energy point.
  • Kinematic cuts applied to match experimental data ($p_T$ and rapidity ranges).

3. Key Contributions

1. Systematic EOS Sensitivity Analysis: This is one of the first studies to systematically investigate how the choice of EOS affects global $\Lambda$ polarization in low-energy transport models, moving beyond hydrodynamic approximations.
2. Identification of the Correct EOS Regime: The study demonstrates that the Hadron Resonance Gas (HRG) EOS is the only scenario capable of reproducing experimental data at low energies, even below the $\Lambda$ production threshold in $NN$ collisions.
3. Prediction of a Polarization Peak: The authors predict a specific peak in global polarization at $\sqrt{s_{NN}} \approx 2.4$ GeV for the HRG EOS, contrasting with peaks at $\approx 3$ GeV found in other EOS scenarios.
4. Helicity Polarization Symmetry: A theoretical demonstration that helicity polarization induced by thermal vorticity vanishes upon azimuthal integration due to space-reversal symmetry in the absence of axial charge.

4. Key Results

• EOS Dependence:

  • HotQCD & NEOS-BQS: Both models underestimate the global polarization at low energies ($\sqrt{s_{NN}} < 7.7$ GeV). They exhibit a non-monotonic trend with a peak around 3 GeV but fail to match the magnitude of STAR and HADES data at 2.4 GeV.
  • HRG Model: The HRG EOS yields the highest polarization values, matching experimental data across the entire low-energy range.
  • Mechanism: The HRG EOS extracts a lower temperature for a given energy density compared to the crossover EOS. Since polarization scales inversely with temperature ($P \propto 1/T$), the lower temperature in the hadronic phase leads to higher thermal vorticity and thus higher polarization.

• Energy Dependence and the Peak:

  • The global polarization increases as energy decreases.
  • Using the HRG EOS, the maximum polarization is predicted to occur around $\sqrt{s_{NN}} = 2.4$ GeV.
  • At $\sqrt{s_{NN}} = 2m_N$ (the kinematic threshold where no collisions occur), vorticity and polarization vanish, consistent with theoretical expectations.

• Centrality, Rapidity, and $p_T$ Dependence (at $\sqrt{s_{NN}} = 3$ GeV):

  • Centrality: Polarization increases with centrality (more peripheral collisions), consistent with experimental trends due to stronger geometric anisotropy and lower freeze-out temperatures.
  • Rapidity ($y_{cm}$): A slight increase in polarization is observed with increasing $|y_{cm}|$.
  • Transverse Momentum ($p_T$): No significant dependence on $p_T$ is observed within the measured range.
  • The HRG simulation results show excellent agreement with STAR data for all these dependencies.

• Helicity Polarization:

  • The study confirms that the rapidity- and $p_T$-dependent helicity polarization induced by thermal vorticity vanishes after integrating over the azimuthal angle due to space-reversal symmetry ($S_h(p) = -S_h(-p)$).
  • Any non-vanishing helicity polarization observed in experiments must therefore arise from other mechanisms (e.g., polarized fragmentation functions), not thermal vorticity.

5. Significance and Implications

• Validation of Thermal Vorticity: The results confirm that thermal vorticity remains the underlying mechanism for global $\Lambda$ polarization even at very low energies, provided the correct (hadronic) EOS is used.
• Matter Characterization: The success of the HRG model implies that at low collision energies ($\sqrt{s_{NN}} \lesssim 3$ GeV), the produced dense matter behaves as a hadron resonance gas rather than a mixed phase or QGP. This highlights the critical importance of finite baryon density effects in low-energy heavy-ion physics.
• Future Experimental Guidance: The prediction of a polarization peak at 2.4 GeV provides a specific target for upcoming experiments (e.g., at the FAIR or NICA facilities) to further constrain the EOS of dense nuclear matter.
• Theoretical Refinement: The work clarifies that discrepancies in previous low-energy studies may stem from using inappropriate EOS (like zero-baryon-density crossover models) rather than a failure of the thermal vorticity mechanism itself.

In conclusion, this paper bridges the gap between transport models and experimental data in the low-energy regime, establishing the HRG EOS as the necessary framework for describing spin polarization in dense, baryon-rich matter.