Boost-invariant and cylindrically symmetric perfect spin hydrodynamics

This paper presents a numerical solution for boost-invariant, cylindrically symmetric perfect spin hydrodynamics with wounded nucleon initial conditions, revealing a unique coupling between azimuthal and longitudinal spin polarization components that induces total polarization via the longitudinal magnetic and azimuthal electric fields.

The Gist

Imagine a high-energy collision between two heavy atomic nuclei (like gold or lead) as a massive, chaotic party where thousands of subatomic particles are created in a split second. For a tiny fraction of a second, these particles don't just bounce around; they form a super-hot, super-dense "soup" called a quark-gluon plasma. This soup behaves like a perfect fluid, flowing and expanding rapidly.

This paper is about building a mathematical model to understand how this fluid moves and, more specifically, how the tiny "spins" (like tiny internal tops) of the particles inside it behave.

Here is a breakdown of the paper's journey, using simple analogies:

1. The Setup: A Symmetrical Explosion

The authors decided to simplify the problem by assuming the explosion happens in a very specific, symmetrical way.

• Boost Invariance: Imagine the fluid expanding like a loaf of bread rising in an oven. No matter where you look along the length of the loaf (the direction the particles are flying), the physics looks the same. It's like stretching a piece of taffy; the pattern repeats itself as it stretches.
• Cylindrical Symmetry: Imagine the explosion looking like a perfect cylinder expanding outward from the center, like a firework shell bursting into a perfect ring. There is no "lopsided" shape; it's the same in every direction around the center.

2. The Two Layers: The Background and the Spin

The authors treat the system in two layers, like a stage play:

• The Background (The Stage): First, they calculate how the fluid itself moves (its temperature, pressure, and flow speed). They ignore the spins for a moment to get the "stage" set. They use a model based on a gas of heavy particles (like protons) to see how this fluid cools down and expands.
• The Spin (The Actors): Once the stage is set, they introduce the "actors"—the spin of the particles. They ask: "If the fluid flows this way, how do the tiny internal tops of the particles spin?"

3. The Big Discovery: A "Handshake" Between Directions

In simpler models (where the fluid only expands in one direction), the different parts of the spin were independent. But in this 2D cylindrical model, the authors found a fascinating connection, or a "handshake," between different directions.

• The Analogy: Imagine a spinning top. Usually, if you push it north, it might tilt north. But in this fluid, if you have a "magnetic" spin pointing up and down (longitudinal), the fluid's expansion forces it to create an "electric" spin pointing sideways (azimuthal, or around the circle).
• The Result: You cannot have one without the other in this specific geometry. The fluid's expansion couples these two different types of spin together. The authors note this is similar to what happens in a different type of symmetry (Gubser symmetry), but they proved it happens even without those strict extra rules.

4. The "Freeze-Out": Taking the Snapshot

Eventually, the hot soup cools down enough that the particles stop interacting and fly off freely. This moment is called "freeze-out."

• The authors define this moment as the point where the temperature hits a specific constant value (like a thermostat turning off).
• They calculate a specific vector called the Pauli-Lubański vector. Think of this as a "spin report card" for the particles as they leave the party. It tells us the average direction the particles are spinning when they are finally detected.

5. The Final Conclusion: What Actually Spins?

After running complex computer simulations with different particle masses, they found a surprising restriction on what can be observed:

• The "Only One" Rule: For this specific cylindrical shape, the only way to get a net spin that points up or down (longitudinal) is if the fluid has a specific combination of "magnetic" spin pointing up/down and "electric" spin pointing sideways.
• The Zeroes: If you start with spins pointing purely radially (outward from the center) or purely in other combinations, they cancel out or don't produce a net observable spin in the final direction.

Summary

In short, the authors built a computer model of a perfectly symmetrical, expanding fluid made of spinning particles. They discovered that the fluid's expansion forces different types of spins to mix together in a specific way. When the fluid finally cools and the particles fly off, the only way to see a net spin pointing along the direction of the collision is if those specific "mixed" spins were present in the fluid.

This work serves as a "reference point" or a clean, controlled test case. It helps physicists understand the basic rules of spin hydrodynamics before they try to apply them to the much messier, real-world collisions that happen in particle accelerators.

Technical Summary

1. Problem Statement

The paper addresses the theoretical development of spin hydrodynamics, a framework designed to describe the evolution of spin polarization in relativistic heavy-ion collisions. While experimental data from collaborations like STAR and ALICE have confirmed global hyperon polarization and vector-meson alignment, the theoretical understanding of spin dynamics in complex geometries remains incomplete.

Previous theoretical works have largely focused on:

• 1D Boost-invariant expansion (Bjorken flow), which assumes transverse homogeneity.
• Gubser symmetry, which imposes conformal symmetry and specific transverse profiles.

The authors identify a gap in understanding 2D cylindrically symmetric systems that allow for general initial conditions (specifically those inspired by the wounded nucleon model) without the restrictive conformal symmetry of Gubser flow. The goal is to solve the equations of perfect spin hydrodynamics in this geometry to understand how radial expansion couples different components of the spin polarization tensor and to calculate observable spin polarization (Pauli–Lubański vector) at freeze-out.

2. Methodology

Theoretical Framework

• Perfect Spin Hydrodynamics: The authors utilize the GLW (de Groot–van Leeuwen–van Weert) formulation. In this scheme, the spin part of the total angular momentum is separately conserved ($\partial_\mu S^{\mu,\alpha\beta} = 0$), implying no exchange between spin and orbital angular momentum in the perfect fluid limit.
• Equation of State (EoS): The system is modeled as a relativistic massive gas governed by Boltzmann statistics. The energy-momentum tensor ($T^{\mu\nu}$) and spin tensor ($S^{\mu,\alpha\beta}$) are expanded to first order in the spin polarization tensor $\omega_{\mu\nu}$.
• Symmetry Implementation: The system assumes boost invariance (along the beam axis $z$) and cylindrical symmetry (in the transverse plane). The fluid four-velocity $U^\mu$ and the spin polarization tensor are decomposed using a basis of four-vectors: $U^\mu$ (time-like), $R^\mu$ (radial), $\Phi^\mu$ (azimuthal), and $Z^\mu$ (longitudinal).

Decomposition of Spin Polarization

The spin polarization tensor $\omega_{\mu\nu}$ is decomposed into "electric-like" ($k_\mu$) and "magnetic-like" ($\omega_\mu$) components orthogonal to the flow. These are further projected onto the radial ($r$), azimuthal ($\phi$), and longitudinal ($z$) directions, yielding six coefficients: $C_{kr}, C_{k\phi}, C_{kz}$ and $C_{\omega r}, C_{\omega \phi}, C_{\omega z}$.

Numerical Approach

1. Background Evolution: The authors first solve the standard hydrodynamic equations for the spinless background (temperature $T$, chemical potential $\mu$, and transverse rapidity $\theta$) using initial conditions based on the Woods-Saxon nuclear density distribution (Wounded Nucleon Model).
2. Spin Evolution: Using the pre-calculated background fields as input, they numerically solve the coupled partial differential equations for the six spin coefficients.
3. Freeze-out: The system freezes out on a hypersurface defined by a constant temperature condition ($T = \text{const}$).
4. Observable Calculation: The mean spin polarization (Pauli–Lubański four-vector) is calculated using the Cooper–Frye formula, integrating over the freeze-out hypersurface. The results are boosted to the particle rest frame (PRF) to obtain physical observables.

3. Key Contributions

• Generalization of Symmetry: Unlike previous studies restricted to Gubser flow (conformal symmetry), this work allows for arbitrary initial transverse profiles (Woods-Saxon) and non-conformal equations of state (massive particles).
• Discovery of Coupling Mechanisms: The authors demonstrate that in 2D cylindrical expansion, the azimuthal and longitudinal components of the spin polarization tensor become coupled. Specifically:
  • The longitudinal magnetic component ($C_{\omega z}$) induces an azimuthal electric component ($C_{k\phi}$).
  • The longitudinal electric component ($C_{kz}$) induces an azimuthal magnetic component ($C_{\omega \phi}$).
  • This coupling is driven by the radial expansion (transverse flow) and is absent in 1D boost-invariant or transversely homogeneous cases.
• Numerical Validation: The code is benchmarked against massless limits and previous 1D results, showing excellent agreement.
• Pauli–Lubański Vector Analysis: The paper provides the first numerical calculation of the full momentum-dependent spin polarization vector for a cylindrically symmetric, boost-invariant system with massive particles.

4. Results

Hydrodynamic Background

• The transverse expansion is faster for lighter particles ($m=0.3$ GeV) compared to heavier ones ($m=1$ GeV) due to a higher speed of sound ($c_s$).
• The temperature and chemical potential profiles evolve smoothly, with the rarefaction wave propagating inward.

Spin Dynamics

• Decoupled Modes: The radial electric ($C_{kr}$) and radial magnetic ($C_{\omega r}$) components evolve independently of the other components.
• Coupled Modes:
  • Case (c): Initializing $C_{\omega z}$ (longitudinal magnetic) dynamically generates a non-zero $C_{k\phi}$ (azimuthal electric). The induced component has an opposite sign.
  • Case (d): Initializing $C_{kz}$ (longitudinal electric) generates a non-zero $C_{\omega \phi}$ (azimuthal magnetic).
  • This behavior mimics Maxwell's equations, where time-varying fields induce orthogonal components.
• Mass Dependence: The evolution of coefficients depends on particle mass, with heavier masses showing different decay/growth rates for the radial components.

Pauli–Lubański Vector (Observables)

• Longitudinal Polarization: The analysis reveals that for the assumed geometry, the only non-zero total polarization in the particle rest frame arises from the coupling of the longitudinal magnetic component ($C_{\omega z}$) with the azimuthal electric component ($C_{k\phi}$).
• Symmetry Constraints:
  • If only radial components are initialized, the total polarization vanishes.
  • The longitudinal component of the Pauli–Lubański vector ($\langle \pi^*_z \rangle$) is cylindrically symmetric in the transverse momentum plane.
  • The transverse components ($\langle \pi^*_x \rangle, \langle \pi^*_y \rangle$) exhibit specific angular dependencies determined by the initial spin configuration.

5. Significance

• Reference for Realistic Models: This work serves as a crucial "reference point" for more complex, fully 3D dissipative spin hydrodynamic simulations. It isolates the effects of geometry and expansion without the noise of full 3D turbulence or viscosity.
• Understanding Spin-Orbit Coupling: The paper clarifies how the geometry of the collision (specifically the radial expansion in a cylindrically symmetric system) generates specific couplings between spin components that were previously unknown or only seen in conformal flows.
• Interpretation of Experiments: The findings suggest that observed spin polarization patterns in heavy-ion collisions may be significantly influenced by the coupling between longitudinal and azimuthal spin components driven by transverse flow. This helps interpret experimental data regarding global vs. local polarization.
• Methodological Advancement: The successful numerical implementation of spin hydrodynamics with massive particles and realistic initial conditions (Woods-Saxon) demonstrates the feasibility of moving beyond idealized massless/conformal models.

In summary, this paper bridges the gap between simple 1D spin hydrodynamics and complex 3D simulations, revealing that radial expansion in cylindrical symmetry inherently couples longitudinal and azimuthal spin degrees of freedom, a feature essential for accurately modeling spin phenomena in relativistic heavy-ion collisions.