Dissipative spin hydrodynamics in Bjorken flow and thermal dilepton production

This paper investigates boost-invariant spin hydrodynamics with a first-order framework, demonstrating that spin dissipation alters the temperature evolution of the expanding medium and subsequently enhances thermal dilepton production, thereby suggesting dileptons as a viable probe for spin dynamics in the quark-gluon plasma.

The Gist

Imagine a heavy-ion collision (like smashing two gold atoms together at nearly the speed of light) as creating a tiny, super-hot drop of "primordial soup" called the Quark-Gluon Plasma (QGP). This soup is so hot that protons and neutrons melt into their constituent parts: quarks and gluons.

For decades, physicists have studied this soup using Hydrodynamics, which is basically the physics of fluids. They treat the soup like a giant, expanding balloon of hot gas. But recently, scientists realized they were missing a crucial ingredient: Spin.

Think of Spin not as the fluid moving, but as the tiny, intrinsic "twirl" or "magnetism" of every single particle inside the soup. In the real world, when you spin a top, it wobbles. In this plasma, the particles are spinning so fast and interacting so chaotically that their collective "twirl" creates a kind of internal whirlpool (vorticity).

Here is what this paper does, explained through simple analogies:

1. The New Rulebook: Adding "Twirl" to the Fluid

Previously, the rulebook for this plasma (Hydrodynamics) only tracked how the fluid moved, how hot it was, and how much pressure it had. It ignored the fact that the particles were spinning.

This paper introduces a Spin Hydrodynamic Framework.

• The Analogy: Imagine a crowd of people running down a hallway. Standard hydrodynamics tracks how fast the crowd moves and how hot the hallway is. This new framework tracks not just the running, but also how much everyone is spinning their arms or twirling as they run.
• The Twist: The authors treat this "spinning" (called the spin chemical potential) as a major player right from the start, not just a tiny side effect. They ask: "If the particles are spinning, how does that change the way the whole crowd moves and cools down?"

2. The Experiment: The "Bjorken Flow" (The Stretching Sausage)

To solve the math, they simplified the scenario to a "Bjorken Flow."

• The Analogy: Imagine a long, hot sausage being stretched out very quickly in one direction (like pulling taffy). The sausage gets thinner and cooler as it stretches.
• The Physics: In heavy-ion collisions, the plasma expands rapidly along the beam line (like the sausage stretching) but stays uniform sideways. The authors used this "stretching sausage" model to see how the "twirl" of the particles evolves as the soup expands.

3. The Discovery: The "Magnetic" vs. "Electric" Twirl

The particles' spin can be thought of in two ways: like a compass needle pointing up/down (Electric-like) or like a spinning top (Magnetic-like).

• The Finding: In this stretching sausage scenario, the "compass needle" parts of the spin vanish instantly. Only the "spinning top" parts survive.
• The Decay: The authors found that the "spinning tops" pointing sideways (transverse) lose their energy and stop spinning very quickly due to friction (dissipation). However, the "spinning top" pointing along the direction of the stretch (longitudinal) keeps spinning for a much longer time. It's like a figure skater spinning on ice; if they spin along their body axis, they stay spinning longer than if they try to wobble sideways.

4. The Big Surprise: Spin Keeps the Soup Hot

This is the most exciting part. The authors discovered that the act of the particles spinning and interacting actually changes how fast the soup cools down.

• The Analogy: Imagine you have a cup of hot coffee. Usually, it cools down steadily. But imagine if the coffee had a magical property where the swirling motion of the liquid generated a little bit of extra heat, slowing down the cooling process.
• The Result: The "spin dynamics" act like that magical property. Because the particles are interacting via their spin, the plasma stays hot for longer than it would if the particles weren't spinning. The "twirl" effectively slows down the cooling of the universe's smallest fire.

5. The Proof: Thermal Dileptons (The Ghost Messengers)

How do we know this is true? We can't stick a thermometer in the plasma. Instead, physicists look for Thermal Dileptons.

• The Analogy: Dileptons are like "ghost messengers." They are pairs of particles (an electron and a positron) created inside the hot soup. Unlike other particles that get stuck in the soup and bounce around, these messengers are "ghosts"—they don't interact with the soup at all. They fly straight out, carrying a perfect memory of how hot the soup was when they were born.
• The Conclusion: Since the "spin" keeps the soup hot for longer, there is more time for these ghost messengers to be born. Therefore, if spin is real and active, we should see more of these messengers coming out of the collision than we would expect from a standard, non-spinning fluid.

Summary

The paper argues that the "twirling" nature of particles in the quark-gluon plasma is not just a detail; it's a fundamental force that:

1. Changes how the plasma expands.
2. Keeps the plasma hot for a longer time.
3. Increases the number of "ghost messengers" (dileptons) we detect.

By measuring these messengers, scientists might finally be able to "see" the spin dynamics of the early universe, proving that the "twirl" of the particles is a key part of the story of how our universe cooled down after the Big Bang.

Technical Summary

1. Problem Statement

Heavy-ion collision experiments have observed significant spin polarization of hadrons (e.g., $\Lambda$ hyperons), attributed to the coupling between particle spin and the vorticity of the quark-gluon plasma (QGP). While standard hydrodynamic models explain global polarization, they struggle to account for local polarization patterns (e.g., longitudinal polarization as a function of azimuthal angle).

Existing spin hydrodynamic frameworks often treat the spin chemical potential ($\omega_{\mu\nu}$) as a first-order gradient term ($O(\partial)$), which is appropriate for asymmetric energy-momentum tensors but theoretically inconsistent for symmetric ones. Furthermore, previous studies of Bjorken flow (boost-invariant expansion) in spin hydrodynamics have largely neglected dissipative effects (spin diffusion) or treated them as non-dynamical.

The core problem addressed: How does a consistent, first-order dissipative spin hydrodynamic framework, where the spin chemical potential is a leading-order ($O(1)$) variable, affect the evolution of the medium's temperature and, consequently, the production of thermal dileptons in a boost-invariant system?

2. Methodology

Theoretical Framework

The authors utilize a recently developed first-order spin hydrodynamic framework (based on Ref. [47]) with the following key features:

• Symmetric Energy-Momentum Tensor: They assume a symmetric $T^{\mu\nu}$, implying the spin tensor $S^{\lambda\mu\nu}$ is separately conserved ($\partial_\lambda S^{\lambda\mu\nu} = 0$).
• Leading-Order Spin Chemical Potential: Unlike previous works where $\omega_{\mu\nu} \sim O(\partial)$, this framework treats $\omega_{\mu\nu}$ as an $O(1)$ hydrodynamic variable alongside temperature ($T$) and flow velocity ($u^\mu$). This allows for the derivation of constitutive relations for dissipative spin currents at the Navier-Stokes level.
• Constitutive Relations: Using entropy current analysis, they derive relations for dissipative currents (shear stress $\pi^{\mu\nu}$, bulk pressure $\Pi$, and spin diffusion currents) involving transport coefficients: shear viscosity ($\eta$), bulk viscosity ($\zeta$), and spin transport coefficients ($\chi_1, \chi_2, \chi_3, \chi_4$).
• Equation of State (EoS): They employ a phenomenological EoS for a baryon-free system where pressure $P(T, \omega_{\mu\nu})$ includes a term quadratic in the spin chemical potential: $P = P_0(T) + P_1(T)\omega_{\mu\nu}\omega^{\mu\nu}$.

System Setup: Bjorken Flow

• The system is modeled as a boost-invariant expansion (Bjorken flow) along the beam axis ($z$).
• Decomposition: The spin chemical potential is decomposed into "electric-like" ($\kappa^\mu$) and "magnetic-like" ($\omega^\mu$) components.
• Constraints: By enforcing the conservation of total angular momentum and the vanishing of the center-of-mass motion for the fire-cylinder, the authors prove that the electric-like components vanish ($\kappa^\mu = 0$). Only the magnetic-like components ($C_{\omega X}, C_{\omega Y}, C_{\omega Z}$) survive.
• Governing Equations: The authors derive a set of coupled first-order differential equations for:
  1. The proper time evolution of temperature $T(\tau)$.
  2. The evolution of the spin chemical potential components $C_{\omega X}(\tau)$, $C_{\omega Y}(\tau)$, and $C_{\omega Z}(\tau)$.

Numerical Simulation

• Initial Conditions: $\tau_0 = 0.5$ fm, $T_0 = 300$ MeV, and initial spin potentials $C_{\omega i} = 80$ MeV.
• Parameters: Two cases of transport coefficients are tested:
  • Case I: $\eta/s_0 = 5/4\pi$, $\chi_s = 10/4\pi$.
  • Case II: $\eta/s_0 = 1/4\pi$, $\chi_s = 3/4\pi$.
  • (Where $\chi_s = (\chi_2 + \chi_3)/S_0(T)$).
• Observables: The resulting temperature profiles are used to calculate the thermal dilepton production rate via quark-antiquark annihilation ($q\bar{q} \to \gamma^* \to \ell^+\ell^-$).

3. Key Contributions

1. Consistent First-Order Theory: The paper rigorously applies a spin hydrodynamic framework where $\omega_{\mu\nu}$ is $O(1)$, demonstrating that this ordering is necessary for a symmetric energy-momentum tensor to derive well-defined dissipative spin currents.
2. Spin Dissipation Dynamics: The authors explicitly derive and solve the evolution equations showing how spin transport coefficients drive the dissipation of spin degrees of freedom.
3. Anisotropic Decay of Spin: They demonstrate a qualitative difference in the decay rates of spin components:
   • Transverse components ($C_{\omega X}, C_{\omega Y}$) decay rapidly due to coupling with spin transport coefficients ($\chi_2, \chi_3$).
   • Longitudinal component ($C_{\omega Z}$) decays much more slowly as it lacks direct dissipative contributions from these coefficients.
4. Spin-Temperature Coupling: The study establishes a feedback loop where spin evolution modifies the temperature profile, which in turn affects the spin evolution.
5. Dilepton Probe: The paper proposes thermal dileptons as an indirect probe of spin transport coefficients, linking the microscopic spin dynamics to a macroscopic observable.

4. Key Results

• Temperature Evolution:
  • The presence of non-zero spin chemical potential leads to a slower cooling of the medium compared to standard dissipative hydrodynamics (where spin is zero).
  • For sufficiently large spin transport coefficients (Case I), the system exhibits a brief initial reheating phase before cooling, analogous to unphysical reheating in relativistic Navier-Stokes equations at early times (due to large gradients).
• Spin Component Evolution:
  • $C_{\omega X}$ and $C_{\omega Y}$ decay significantly faster than $C_{\omega Z}$.
  • The decay rate of all components is indirectly influenced by the shear viscosity ($\eta/s_0$) via its effect on the temperature evolution.
• Dilepton Production:
  • Enhancement: The slower cooling in spin hydrodynamics extends the lifetime of the hot partonic phase, leading to a significant enhancement in the thermal dilepton yield compared to standard hydrodynamics.
  • Sensitivity: The magnitude of this enhancement depends on the spin transport coefficients.
  • Component Dependence: When only the longitudinal spin component is non-zero, the cooling is faster (shorter lifetime) than when transverse components are present, resulting in a lower dilepton yield. This highlights the specific role of transverse spin dissipation in prolonging the plasma phase.
  • Contrast with Vorticity: Unlike previous studies where finite vorticity reduced dilepton rates, this dynamic spin hydrodynamic approach shows an increase in rates.

5. Significance and Outlook

• Phenomenological Impact: The results suggest that thermal dileptons are a sensitive probe for spin dynamics in the QGP. The enhancement in dilepton yield provides a potential experimental signature to constrain spin transport coefficients ($\chi_s$).
• Theoretical Advancement: The work resolves theoretical inconsistencies regarding the hydrodynamic ordering of the spin chemical potential for symmetric energy-momentum tensors and provides a complete set of dissipative equations for Bjorken flow.
• Future Directions: The authors note that future work should incorporate:
  • Microscopic calculations for spin transport coefficients (currently treated as phenomenological parameters).
  • Lattice QCD equations of state.
  • Finite baryon density and transverse expansion.
  • Application to other observables like thermal photon production.

In summary, this paper demonstrates that spin dynamics are not merely a passive feature of the QGP but actively modify the thermodynamic evolution of the medium, leaving a distinct imprint on electromagnetic probes like dileptons.