Modeling $\Lambda$ polarization in Au$+$Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV using relativistic spin hydrodynamics

This paper employs a novel $(1+1+2)$D ideal relativistic spin hydrodynamics model incorporating transverse flow and longitudinal spin acceleration to successfully reproduce experimental $\Lambda$ hyperon polarization data in Au+Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV and predicts the yet-unmeasured in-plane transverse spin polarization.

The Gist

Imagine a high-energy physics experiment as a massive, high-speed crash between two heavy atomic nuclei (like gold atoms). When these nuclei smash into each other at nearly the speed of light, they create a tiny, super-hot drop of "primordial soup" called the Quark-Gluon Plasma (QGP). This soup is so hot and dense that it behaves like a nearly perfect fluid, swirling and expanding with incredible speed.

This paper is about trying to understand how the tiny particles inside this soup (specifically, particles called Lambda hyperons) end up spinning in a specific direction.

Here is the breakdown of what the authors did, using simple analogies:

1. The Big Picture: The Spinning Ball of Dough

When two gold nuclei collide, they don't just hit head-on; they usually graze each other. Imagine two spinning balls of dough hitting each other sideways. Because they miss the center, the resulting "dough" (the QGP) has a huge amount of orbital angular momentum—it's spinning like a giant, chaotic top.

The big question the scientists wanted to answer is: How does this giant, macroscopic spin get transferred down to the microscopic spin of the individual particles inside? It's like asking how the spin of a giant whirlpool makes the individual water molecules inside it rotate.

2. The Old Map vs. The New Map

To study this, scientists use a set of rules called "hydrodynamics" (the study of fluids).

• The Old Map (Boost-Invariant): Previous models assumed the fluid expanded perfectly symmetrically, like a cylinder stretching out evenly in all directions. It was a simple, flat map.
• The Problem: This simple map couldn't explain everything the experiments saw. Specifically, it failed to explain a specific "four-leaf clover" pattern (called a quadrupole) in how the particles were spinning along the direction of the beam.
• The New Map (Non-Boost-Invariant): The authors created a more realistic map. They realized the fluid doesn't just stretch evenly; it has bumps, dips, and different speeds depending on where you look. They used a sophisticated mathematical solution (the "SJG flow") that allows the fluid to expand in a more complex, realistic way, similar to how a real explosion isn't perfectly uniform.

3. The Two-Step Experiment

The authors ran their simulation in two stages to see what was missing:

Stage 1: The 1D Highway (The (1+1)D Model)
They first looked at the collision as if it were a one-dimensional highway. The fluid could move forward and backward, but they ignored the side-to-side movement.

• Result: This model was good at predicting the average spin of the particles. It told them, "Yes, the particles are spinning in the right direction overall."
• Failure: However, it couldn't explain the local details. It was like knowing the average wind speed in a city but not knowing why the wind is swirling in a specific alleyway. It missed the "four-leaf clover" pattern.

Stage 2: The 3D Explosion (The 1-1-2 Model)
To fix this, they added the missing piece: Transverse Flow. They kept their realistic forward/backward expansion but added a "freeze-out" layer that accounted for the fluid expanding sideways and being squashed into an oval shape (like a flattened football) rather than a perfect circle.

• The Secret Ingredient: They discovered that to get the correct "four-leaf clover" pattern, they needed to include a specific type of "spin acceleration."
• The Analogy: Imagine a figure skater spinning. If they just spin, they have rotation. But if they also push off the ice with their feet while spinning, that "acceleration" changes how their body twists. The authors found that this "spin acceleration" combined with the sideways expansion of the fluid creates the specific pattern seen in the data.

4. The Results

By combining the realistic forward expansion with the sideways "squashing" and the "spin acceleration," their model finally matched the experimental data from the STAR experiment at the Relativistic Heavy Ion Collider (RHIC).

• Global Polarization: They correctly predicted the overall spin direction.
• Local Polarization: They correctly predicted the complex "four-leaf clover" pattern of spin along the beam direction.
• A New Prediction: The model also predicted a specific type of spin polarization that happens sideways (in the plane of the collision). The authors note that, as far as they know, no one has measured this specific sideways spin yet. It's like predicting a new flavor of ice cream that hasn't been tasted by anyone.

Summary

The paper is essentially a story about upgrading a weather forecast model.

1. Old Model: "It's windy." (Too simple, misses the details).
2. New Model: "It's windy, but the wind swirls differently depending on the shape of the buildings and the acceleration of the air."
3. Outcome: The new model perfectly predicts the wind patterns (spin polarization) observed in the lab.

The authors conclude that to understand how particles spin in these high-energy crashes, we cannot just look at the big picture; we must account for the complex, uneven way the fluid expands and the specific "acceleration" forces acting on the spins. They have provided a mathematical toolkit that successfully explains the data and offers a new prediction for future experiments to test.

Technical Summary

Technical Summary: Modeling $\Lambda$ Polarization in Au+Au Collisions at $\sqrt{s_{NN}} = 200$ GeV using Relativistic Spin Hydrodynamics

Problem Statement
Relativistic heavy-ion collisions generate a quark-gluon plasma (QGP) characterized by extreme temperature, energy density, and significant initial orbital angular momentum. A central question in the field is how this macroscopic angular momentum is redistributed and converted into the microscopic spin polarization of produced hadrons, particularly $\Lambda$ hyperons. While global polarization has been successfully measured, describing the local polarization structure—specifically the azimuthal dependence and the longitudinal component—remains challenging. Previous theoretical attempts, often relying on boost-invariant (Bjorken) backgrounds or simple thermal vorticity assumptions, have struggled to reproduce the observed quadrupole structure in the longitudinal polarization and have sometimes predicted incorrect signs or magnitudes compared to experimental data from the STAR collaboration. Furthermore, the role of spin acceleration and the interplay between longitudinal non-boost-invariant dynamics and transverse anisotropy require further investigation within a consistent hydrodynamic framework.

Methodology
The authors investigate spin polarization dynamics using ideal relativistic spin hydrodynamics within the de Groot-van Leeuwen-van Weert (GLW) pseudogauge formulation. The study operates in the small-polarization regime, where the spin potential $\omega_{\mu\nu}$ is treated perturbatively. This allows the bulk hydrodynamic evolution to be solved independently of the spin sector, with the spin degrees of freedom evolving on top of this background.

The methodology proceeds in two stages:

1. (1+1)D Setup: The authors employ exact, non-boost-invariant longitudinal solutions for ideal relativistic hydrodynamics (the "SJG flow" from Shi, Jeon, and Gale). This background captures realistic rapidity dependence in temperature and flow velocity without assuming boost invariance. The system is assumed to be homogeneous in the transverse plane. The spin potential components are initialized based on symmetry constraints derived from the conservation of total angular momentum in non-central collisions.
2. (1+1+2)D Extension: To address the limitations of the (1+1)D model in reproducing azimuthal structures, the authors construct a phenomenological extension. While the longitudinal dynamics remain governed by the exact SJG solution, transverse flow and spatial anisotropy (elliptic deformation) are incorporated at the freeze-out hypersurface. This "1-1-2" model allows for the inclusion of transverse flow components ($u_x, u_y$) and a deformed freeze-out geometry without solving the full (3+1)D spin hydrodynamic equations.

Spin polarization observables for $\Lambda$ hyperons are computed using the Cooper-Frye freeze-out procedure and the GLW expression for the spin polarization vector, which depends on the dual spin potential and the particle momentum.

Key Contributions

• Non-Boost-Invariant Background: The application of exact non-boost-invariant longitudinal solutions to spin hydrodynamics, moving beyond the restrictive Bjorken limit to better model rapidity-dependent spin evolution.
• Symmetry-Constrained Initialization: A rigorous derivation of the necessary symmetry properties (even/odd in spacetime rapidity $\eta_s$) for the spin potential components to ensure the global polarization aligns with the total angular momentum (along the $-\hat{y}$ direction).
• Identification of Spin Acceleration: The demonstration that a longitudinal spin acceleration component ($a_z$), coupled with transverse expansion, is essential for generating the observed quadrupole pattern in the longitudinal polarization ($P_z$).
• Minimal Model for Quadrupole Structure: The construction of a minimal (1+1+2)D framework that successfully reproduces the azimuthal modulation of local longitudinal polarization, a feature that purely (1+1)D models fail to capture due to transverse homogeneity.

Results

• (1+1)D Results: The (1+1)D model successfully reproduces the qualitative features of global polarization and the sign of the in-plane transverse polarization ($P_x$). It generates a quadrupole pattern in $P_x$ driven by directed flow. However, due to transverse homogeneity, it fails to produce the experimentally observed quadrupole structure in the longitudinal polarization ($P_z$), yielding a vanishing second Fourier harmonic $\langle P_z \sin(2\phi) \rangle$.
• (1+1+2)D Results: The inclusion of transverse flow and elliptic freeze-out geometry, combined with a non-vanishing longitudinal spin acceleration ($a_z$), leads to the emergence of a quadrupole pattern in $P_z$. The model achieves qualitative and reasonably good quantitative agreement with experimental data for:
  • Global polarization ($\langle P_J \rangle$) as a function of centrality and rapidity.
  • Local longitudinal polarization ($P_z$) as a function of azimuthal angle and transverse momentum ($p_T$).
  • In-plane transverse polarization ($P_x$) and its Fourier coefficients.
• Momentum Dependence: The model predicts that the second harmonic of the longitudinal polarization, $\langle P_z \sin(2\phi) \rangle$, increases at low $p_T$, reaches a maximum, and then decreases, contrasting with the linear growth seen in in-plane polarization. This behavior is attributed to the specific coupling of $a_z$ with the transverse volume element in the freeze-out integral.
• New Prediction: The paper provides predictions for the in-plane transverse spin polarization ($P_x$) as a function of azimuthal angle and $p_T$, an observable noted as not yet experimentally measured.

Significance and Claims
The paper claims that ideal relativistic spin hydrodynamics, when formulated with non-boost-invariant longitudinal dynamics and supplemented by a phenomenological transverse freeze-out geometry, provides a viable framework for understanding spin polarization in high-energy Au+Au collisions. The work identifies a minimal mechanism—the interplay between longitudinal spin acceleration and transverse expansion—sufficient to explain the complex azimuthal structures observed in data without requiring full (3+1)D dissipative simulations.

The authors emphasize that their results support the hypothesis that spin degrees of freedom reach equilibration at early times ($\tau_s \approx 1$ fm/c) and that the spin potential initialization, constrained by symmetry and angular momentum conservation, is sufficient to describe the data. They conclude that while dissipation and gradient corrections may be subleading in high-energy central collisions, the inclusion of acceleration effects is crucial for accurate phenomenology. The study underscores the necessity of moving beyond highly symmetric setups to fully characterize the space-time structure of the QGP via spin observables.