Excitation function for global \Lambda polarization in relativistic heavy ion collisions with the Core Corona model

This paper presents a Core-Corona model incorporating intrinsic polarization functions derived from a field-theoretical approach with effective mediators, which successfully describes the excitation function of global $\Lambda$ polarization across the full experimental energy range by highlighting the dominant role of the dilute corona and predicting a robust maximum near $\sqrt{s_{NN}} \sim 3$ GeV.

The Gist

Imagine two heavy trucks crashing into each other at nearly the speed of light. In the world of particle physics, these "trucks" are heavy atomic nuclei (like gold), and the crash happens inside a giant accelerator. When they collide, they don't just bounce off; they smash together to create a tiny, super-hot drop of "primordial soup" called Quark-Gluon Plasma (QGP). This is the state of matter that existed just microseconds after the Big Bang.

One of the strangest things scientists have discovered in these crashes is that the particles flying out of the wreckage (specifically a type of particle called a Lambda hyperon) seem to be "spinning" in a specific direction. It's as if the entire explosion had a giant, invisible whirlpool inside it, and the particles caught in the current started spinning like tops.

This paper is about figuring out why these particles spin and how much they spin at different crash speeds.

The Core Idea: The "Core" and the "Corona"

The authors use a clever model called the Core-Corona framework to explain what happens inside the crash. Think of the collision zone like a storm:

1. The Core (The Eye of the Storm): In the very center of the crash, the density is so high that the matter melts into a fluid-like soup (the QGP). This is the "Core." It's hot, dense, and chaotic.
2. The Corona (The Outer Cloud): Around the edges, the matter isn't dense enough to melt into a soup. It's more like a dilute gas where individual particles just bounce off each other, similar to how two cars might graze each other in a traffic jam. This is the "Corona."

The paper argues that to understand the spinning particles, you have to look at both the Core and the Corona, not just the center.

The "Whirlpool" Effect (Vorticity)

When two heavy trucks crash, they don't just hit head-on; they usually miss slightly. This creates a massive amount of angular momentum (spin). Imagine two figure skaters colliding while spinning; the whole system starts to rotate.

In the collision, this rotation creates a vorticity—a giant, microscopic whirlpool. The paper suggests that the particles (Lambdas) get caught in this whirlpool. Just like a leaf caught in a river current will spin along with the water, these particles align their spin with the rotation of the medium.

The New Twist: The "Corona" is the Star

Previous theories focused mostly on the Core (the hot soup) to explain this spinning. However, this paper introduces a major update: The Corona is actually doing most of the work.

Here is the analogy:

• Imagine you are trying to spin a coin on a table.
• The Core is like a thick, sticky syrup. It's hard to move through, and the coin spins slowly because the syrup resists it.
• The Corona is like a thin layer of oil. It's easier to move through, and the coin can spin faster and align better with the current.

The authors found that at lower collision energies (slower crashes), the "Corona" (the outer, less dense region) lives longer and is bigger. Because it lasts longer, the particles have more time to align their spin with the whirlpool. This explains why the spinning effect is actually stronger at lower energies than scientists previously thought.

The "Magic Formula" (The Math Part)

To prove this, the authors did some heavy lifting with math (which they call "field theory").

• They created a new mathematical tool (a "propagator") to describe how particles move when they are in a spinning environment. Think of this as a map that shows how a particle behaves when the whole room is rotating.
• They calculated how long the "Core" and "Corona" last (their "lifetime").
• They combined these numbers to predict exactly how much the particles should spin at different crash speeds.

The Results: A Perfect Match

When they compared their calculations to real data from experiments (like those at the STAR detector), the results were amazing:

• The Curve: The model predicted that as you slow down the collision energy, the spinning effect gets stronger, reaches a peak around a specific energy (3 GeV), and then drops off.
• The Fix: Earlier models failed to explain data from very low-energy crashes (like those from the HADES experiment). By allowing for a "sub-threshold" production (basically saying particles can be made even when the energy seems too low in a nuclear environment), the new model fixed this problem.
• The Prediction: The model predicts a "sweet spot" or maximum spinning effect near 3 GeV. This is a robust prediction that future experiments can test.

Why Does This Matter?

Understanding this "spinning" helps us understand the viscosity (stickiness) of the primordial soup created in the Big Bang. It tells us how the universe behaved in its first fractions of a second.

In summary:
This paper is like a detective story. The mystery was: "Why do particles spin in heavy-ion collisions, and why does the spinning change with speed?" The solution was realizing that the "outer edges" of the crash (the Corona) are just as important as the "center" (the Core). By accounting for the outer edges and how long they last, the authors built a model that perfectly explains the data and predicts a new peak in the spinning behavior that we can look for in future experiments.

Technical Summary

1. Problem Statement

Relativistic heavy-ion collisions generate extreme conditions of temperature and density, creating a medium with intense vortical motion. Experimental observations (e.g., by the STAR collaboration) have revealed a global polarization of $\Lambda$ hyperons that increases as the center-of-mass collision energy ($\sqrt{s_{NN}}$) decreases. However, existing theoretical models struggle to describe the full excitation function across the entire energy range (from $\sim 2$ GeV to 200 GeV), particularly:

• Failing to describe low-energy data points (e.g., HADES data) where $\Lambda$ production is near or below the nucleon-nucleon threshold.
• Accurately accounting for the relative contributions of the dense Quark-Gluon Plasma (QGP) core versus the dilute hadronic corona.
• Providing a unified field-theoretical framework that incorporates the effects of rotation on fermion propagators in both regions.

2. Methodology

The authors employ a Core–Corona framework combined with a field-theoretical approach to calculate the global $\Lambda$ polarization.

A. The Core–Corona Model

The interaction region in semi-central collisions is divided into two distinct subregions based on participant density ($n_p$):

• The Core: A high-density region ($n_p > n_c$, where $n_c = 3.5 \text{ fm}^{-2}$) where a thermalized QGP is formed. $\Lambda$ production here is driven by coalescence of strange quarks.
• The Corona: A dilute outer region ($n_p < n_c$) where interactions resemble proton-proton collisions. $\Lambda$ production here occurs via hadronic recombination (REC).
• Global Polarization ($P_\Lambda$): Calculated as a weighted sum of the intrinsic polarizations ($z$) of the core and corona, weighted by their respective $\Lambda$ yields ($N_\Lambda$):
  $$P_\Lambda = \frac{N_\Lambda^{QGP}}{N_\Lambda^{total}} z_{QGP} + \frac{N_\Lambda^{REC}}{N_\Lambda^{total}} z_{REC}$$

B. Field-Theoretical Derivation of the Fermion Propagator

A key theoretical innovation is the derivation of an effective fermion propagator in a rotating environment.

• Geometry: The medium is modeled as a rigid cylinder rotating with constant angular velocity $\Omega$ around the $\hat{z}$-axis.
• Metric: The rotating frame is described by an effective metric tensor resembling curved spacetime.
• Derivation: Using the Fock-Schwinger proper-time method, the authors solve the Dirac equation in this rotating metric. They derive a compact expression for the propagator $S(p)$ in momentum space, which includes spin-projection operators ($O_\pm$) and energy shifts due to rotation ($\pm \Omega/2$).
  $$S(p) = \frac{\gamma_0(p_0 + \Omega/2) - \vec{\gamma}\cdot\vec{p} + m}{(p_0 + \Omega/2)^2 - p^2 - m^2 + i\epsilon} O_+ + \dots (\text{for } O_-)$$
• Application: This propagator is used to compute the self-energy of $\Lambda$ hyperons and strange quarks, encoding the vortical motion of the medium.

C. Interaction Rates and Relaxation Times

The intrinsic polarization ($z$) in each region is determined by the relaxation time ($\tau$) required for spin to align with vorticity:
$$z = 1 - e^{-\Delta\tau / \tau}$$

• Core (QGP): Interactions are mediated by gluons (using HTL approximation). The relaxation time is derived from the imaginary part of the strange quark self-energy.
• Corona (Hadronic): Interactions are mediated by $\sigma$-mesons (isoscalar-scalar) using a Relativistic Mean Field (RMF) approach. The relaxation time is derived from the $\Lambda$ self-energy involving the effective $\sigma$-propagator at finite temperature and baryon chemical potential.

D. Kinematic and Thermodynamic Inputs

• Yields: $\Lambda$ yields in the core and corona are estimated using a Glauber model with a critical density threshold.
• Subthreshold Production: To describe low-energy data, the model allows the $\Lambda$ production cross-section ($\sigma_{NN}^\Lambda$) to be non-zero below the standard nucleon-nucleon threshold, effectively shifting the threshold to $\sqrt{s_{NN}} \approx 2.1$ GeV.
• Thermodynamics: Temperature ($T$) and baryon chemical potential ($\mu_B$) are fixed along the chemical freeze-out curve.
• Lifetimes: The model parametrizes the lifetimes ($\Delta\tau$) and volumes of the core and corona, noting that the corona volume becomes dominant at lower energies due to nuclear stopping.

3. Key Contributions

1. Explicit Propagator Derivation: The paper provides a rigorous derivation of the fermion propagator in a rotating frame, correcting and extending previous approximations. This allows for a consistent calculation of spin-vorticity alignment rates.
2. Unified Core-Corona Description: It successfully integrates QGP (gluon-mediated) and hadronic (meson-mediated) mechanisms into a single framework, demonstrating that the corona contribution is dominant for the centralities corresponding to experimental data.
3. Subthreshold Mechanism: The introduction of a lowered production threshold for $\Lambda$ in the nuclear environment resolves the discrepancy with low-energy HADES data, which previous models could not describe.
4. Robust Prediction: The model predicts a specific shape for the excitation function, including a distinct maximum, which is shown to be stable under variations of freeze-out parameters.

4. Results

• Excitation Function: The calculated global $\Lambda$ polarization excitation function shows excellent agreement with experimental data from STAR (Au+Au) and HADES (Ag+Ag) across the range $\sqrt{s_{NN}} = 2.55$ GeV to $200$ GeV.
• Dominance of the Corona: For the centralities where data exists (20–50%), the polarization is found to originate primarily from the corona region, not the QGP core.
• The Maximum: The model predicts a robust maximum in global polarization near $\sqrt{s_{NN}} \approx 3$ GeV.
  • Below this energy, polarization drops rapidly as the system approaches the production threshold.
  • Above this energy, polarization decreases as the collision energy increases.
• Centrality Dependence: The model reproduces the observed trend where polarization increases with centrality (impact parameter) for fixed energies.
• Sensitivity: The position of the maximum is sensitive to the baryon chemical potential ($\mu_B$) but remains stable under reasonable variations of the freeze-out curve and the subthreshold cross-section.

5. Significance

• Theoretical Validation: The work validates the hypothesis that global hyperon polarization is a direct consequence of the vortical motion of the medium, successfully linking microscopic field-theoretical calculations to macroscopic experimental observables.
• Low-Energy Physics: By accounting for subthreshold production and the dominance of the hadronic corona, the model provides a consistent explanation for data in the low-energy regime (NICA/FAIR energy range), which is crucial for future experiments like MPD at NICA and CBM at FAIR.
• Methodological Advancement: The explicit derivation of the rotating fermion propagator offers a new tool for studying spin dynamics in rotating quantum field theories, applicable beyond heavy-ion collisions (e.g., neutron stars).
• Predictive Power: The prediction of a peak near 3 GeV serves as a critical benchmark for upcoming experimental campaigns, guiding the search for the maximum vorticity and understanding the transition between hadronic and partonic degrees of freedom.