Late-time attractors in relativistic spin hydrodynamics in Gubser flow

This paper investigates the late-time asymptotic behavior and attractor structure of spin density in minimal causal spin hydrodynamics under Gubser flow, demonstrating that in specific regimes, spin density decays as a hydrodynamic mode governed by late-time scaling laws similar to conventional thermodynamic variables.

The Gist

Imagine a massive, chaotic explosion, like a supernova or a high-speed collision of two heavy nuclei in a particle accelerator. Inside this explosion, a "soup" of subatomic particles is created. Usually, physicists describe this soup using hydrodynamics, treating it like a fluid (like water or honey) that flows, expands, and cools down.

But there's a twist: these particles aren't just moving; they are also spinning (like tiny tops). This paper investigates what happens to that spin as the fluid expands and cools over time.

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Setting: The Expanding Balloon

The authors are studying a specific type of explosion called Gubser flow.

• The Analogy: Imagine a balloon being inflated. It expands in all directions (radially) but also stretches out along a specific line (longitudinally).
• The Problem: In standard physics, we know how the pressure and temperature of the gas inside the balloon change as it expands. But what happens to the spin of the particles? Does the spin disappear quickly, or does it stick around?

2. The Mystery: The "Ghost" of Spin

For a long time, physicists thought that spin was a "fast-decaying" quantity.

• The Analogy: Think of a spinning top on a table. If you spin it, it wobbles and stops very quickly due to friction. The authors suspected that in the particle soup, the spin would act like that top—damping out (disappearing) exponentially fast, leaving no trace by the time the experiment is over.
• The Surprise: This paper asks, "What if the spin doesn't stop? What if it finds a way to survive?"

3. The Discovery: The "Magnetic Track" (Attractors)

The authors used complex math to simulate the evolution of this spinning fluid. They discovered something fascinating called Late-time Attractors.

• The Analogy: Imagine a marble rolling down a hilly landscape with many valleys and peaks.
  • Repellers (Peaks): If you balance the marble perfectly on a sharp peak, it stays there for a moment. But the slightest breeze (a tiny change in conditions) knocks it off, and it rolls away. This is an unstable state.
  • Attractors (Valleys): No matter where you drop the marble on the hill, it eventually rolls down into the deepest valley and settles there.
• The Finding: The authors found that the spin density of the particles has a "valley" it naturally rolls into. Even if the spin starts in a weird state, as time goes on, it gets "trapped" in a specific pattern of behavior. It doesn't vanish; it settles into a predictable rhythm.

4. The Two Outcomes: Fading vs. Flowing

The paper identifies two main ways the spin behaves in this "valley," depending on the properties of the fluid:

Case A: The Slow Fade (Power-Law Decay)

In some scenarios, the spin doesn't disappear instantly. Instead, it fades away slowly, like a candle burning down rather than being snuffed out.

• The Analogy: Imagine a drop of ink in a river. If the river flows fast, the ink spreads out and thins, but it's still there, just very diluted.
• Why it matters: This "slow fade" means the spin is still present when the particles freeze out (stop interacting). This could explain why experiments see more spin polarization than expected. The spin survives long enough to be measured!

Case B: The "Hydrodynamic Mode"

In a very specific, rare set of conditions, the spin stops acting like a spinning top that stops, and starts acting like the fluid itself.

• The Analogy: Usually, a leaf floating on a river is just a passenger. But in this special case, the leaf becomes part of the current. It moves exactly as the water moves.
• The Result: The spin density becomes a "hydrodynamic mode." It decays at the exact same rate as the temperature or pressure of the fluid. It is no longer a "dying" variable; it is a fundamental part of the flow.

5. Why This Matters

This research solves a puzzle in high-energy physics.

• The Puzzle: Experiments at facilities like the Large Hadron Collider (LHC) and RHIC see particles with specific spins. Standard models said the spin should have vanished long ago.
• The Solution: This paper shows that under the right conditions, the spin has a "safety net" (the attractor). It doesn't vanish; it evolves slowly, following the expansion of the universe created in the collision.

Summary

Think of the particle soup as a giant, spinning dance floor.

1. Old View: The dancers (particles) spin wildly at first, but they quickly get tired and stop spinning.
2. New View (This Paper): The dancers find a rhythm. Even as the dance floor expands and the music slows, they don't stop spinning. Instead, they lock into a specific, stable pattern (the attractor) that allows them to keep spinning in sync with the expansion of the floor.

This means that the "spin" of the universe created in these collisions is more persistent and important than we previously thought, potentially leaving a lasting fingerprint on the particles we detect today.

Technical Summary

1. Problem Statement

The paper addresses a critical gap in the understanding of relativistic spin hydrodynamics, specifically regarding the dynamical evolution of spin density in the medium produced in non-central heavy-ion collisions.

• Context: While global polarization of hyperons is well-described by thermal vorticity, local polarization remains challenging to model. Spin hydrodynamics has been proposed as a macroscopic framework to describe spin degrees of freedom.
• The Issue: In canonical spin hydrodynamics, spin density is not a conserved quantity (unlike charge or energy-momentum). Previous analytic solutions in Bjorken flow suggested that spin density relaxes exponentially fast, making it negligible at late times (freeze-out).
• The Question: Does this rapid decay hold in more realistic scenarios involving transverse expansion? Specifically, do late-time attractors exist in Gubser flow (which includes transverse expansion) that allow spin density to persist as a hydrodynamic mode with power-law decay, similar to conventional thermodynamic variables?

2. Methodology

The authors employ a combination of analytical derivation, asymptotic analysis, and numerical simulation within the framework of minimal causal relativistic spin hydrodynamics.

• Theoretical Framework:

  • They utilize the canonical pseudo-gauge formulation of spin hydrodynamics.
  • They adopt a minimal causal theory (second-order gradient expansion) to ensure causality and stability, avoiding the acausality of first-order theories.
  • Constitutive relations for the spin-related heat flux ($q^\mu$) and antisymmetric tensor ($\phi^{\mu\nu}$) are derived, including a specific coupling term ($\lambda_\theta$) to restore conformal covariance.

• Geometric Setup (Gubser Flow):

  • The system is modeled using Gubser flow, which preserves boost invariance along the longitudinal direction while incorporating transverse expansion.
  • To simplify the equations, the authors perform a Weyl transformation mapping the Minkowski spacetime ($R^{3,1}$) to a conformally related spacetime ($dS_3 \times R$).
  • In this transformed frame, the fluid velocity is static ($\hat{u}^\mu = (1,0,0,0)$), and the problem reduces to solving differential equations with respect to the radial coordinate $\rho$ (related to proper time $\tau$).

• Mathematical Analysis:

  • Reduction: The authors reduce the full set of hydrodynamic equations to a single second-order differential equation governing the spin density component $\hat{S}$ (specifically $\hat{S}_{\phi\eta}$).
  • Variable Transformation: They introduce an auxiliary function $f(w)$ and a variable $w$ (related to $\tanh \rho$) to transform the problem into a Riccati-type equation.
  • Asymptotic Expansion: Using the slow-roll expansion method, they derive asymptotic solutions for $f(w)$ in the limit of large $w$ (late proper time).
  • Parameterization: Transport coefficients (relaxation time $\hat{\tau}_\phi$ and coupling $\hat{\gamma}_s$) are parameterized as power laws of $w$ ($w^{\Delta_1}, w^{\Delta_2}$) to explore different physical regimes.

3. Key Contributions

1. Derivation of Governing Equations: The paper derives the specific differential equation for spin density in Gubser flow within a causal framework, explicitly showing how the Weyl transformation affects the constitutive relations.
2. Identification of Attractors and Repellers: By analyzing the Riccati equation, the authors identify distinct attractor (stable) and repeller (unstable) branches for the spin density evolution.
3. Discovery of Power-Law Decay Regimes: Contrary to the exponential decay found in some previous Bjorken flow studies, this work identifies specific parameter regions where spin density decays as a power law ($\tau^{-n}$).
4. Mapping to Physical Space: The asymptotic solutions in the conformal frame are mapped back to flat Minkowski space to determine the physical scaling of spin density components ($S_{tx}, S_{ty}, \dots$) with proper time $\tau$ and system size $L$.

4. Key Results

The behavior of the spin density depends heavily on the scaling exponents $\Delta_1$ and $\Delta_2$ of the transport coefficients:

• Attractor Structure:

  • Numerical simulations confirm that for generic initial conditions, the system converges to a unique attractor branch at late times.
  • Solutions starting on the repeller branch are unstable and diverge from it, eventually settling onto the attractor.
  • Trajectories where $f(w) \to -\infty$ correspond to the spin density crossing zero, after which the solution re-enters from $+\infty$ and approaches the attractor.

• Decay Behaviors:

  • Exponential Decay: In many parameter regimes, $f(w)$ scales such that $\hat{S}$ decays exponentially. This implies spin effects vanish rapidly and are negligible at freeze-out.
  • Power-Law Decay (Case i): When $\Delta_1 > \Delta_2$, the spin density follows $\hat{S} \propto w^{-3/2}$. In the limit where the system size $L \gg \tau$ (early/mid-time), the physical spin density scales as $S \sim \tau^{-2} L^{-3}$. This decay rate is comparable to the charge number density in Bjorken flow ($\sim \tau^{-1}$), suggesting spin can persist as a hydrodynamic mode.
  • Power-Law Decay (Case ii): When $\Delta_1 = \Delta_2$, the decay rate depends on the ratio of transport coefficients $\beta/\alpha$.
    • If $\beta > \alpha$, the spin density can even increase or remain constant with proper time in the $L \gg \tau$ limit. This occurs because the source term (orbital-to-spin angular momentum transfer) decays slower than the expansion-driven dilution.
    • If $\beta < \alpha$, the spin density follows a standard power-law decay.

• Physical Interpretation:

  • The persistence of spin density is driven by a balance between the expansion-driven dilution and the spin-orbit coupling (source term).
  • When the relaxation time is short enough and the coupling to the orbital angular momentum is strong, the spin density is "fed" by the orbital sector fast enough to counteract dilution, behaving as a hydrodynamic mode rather than a rapidly damped non-hydrodynamic variable.

5. Significance

• Phenomenological Impact: The results suggest that spin polarization effects may be observable at late times (freeze-out) in heavy-ion collisions, challenging the assumption that spin relaxes too quickly to matter. This is crucial for interpreting experimental data on hyperon polarization (e.g., from STAR and CMS).
• Theoretical Advancement: The work demonstrates that Gubser flow (with transverse expansion) supports attractor solutions that allow for slow, power-law decay of spin density, a feature not captured in simpler Bjorken flow models.
• Hydrodynamic Mode: It establishes that under specific conditions, spin density behaves as a genuine hydrodynamic mode governed by the late-time scaling laws of the flow, rather than a transient non-hydrodynamic variable.
• Future Directions: The identification of parameter regions where spin density grows or remains constant provides a target for future microscopic calculations (kinetic theory) to constrain transport coefficients ($\gamma_s, \tau_\phi$) and validate the hydrodynamic description of spin.

In summary, the paper provides a rigorous mathematical proof that spin density in relativistic heavy-ion collisions can exhibit attractor behavior leading to power-law decay, thereby remaining relevant for observables at the freeze-out stage, provided the transport coefficients satisfy specific scaling relations.