Friction terms in multi-fluid description of heavy-ion collisions

This paper introduces a novel "charge transfer" friction term within the MUFFIN multi-fluid model for heavy-ion collisions, demonstrating that this approach offers a more consistent description of momentum-separated fluids and improved agreement with experimental data compared to existing methods.

The Gist

Imagine two massive trucks speeding toward each other on a highway, but instead of crashing into a pile of scrap metal, they smash together and create a chaotic, super-hot soup of particles. This is what happens in a heavy-ion collision inside a particle accelerator like the Large Hadron Collider (LHC).

Physicists want to understand the rules of this "soup" (called the Quark-Gluon Plasma) because it existed just after the Big Bang. To do this, they use computer simulations.

This paper is about improving the rules of the road for how these two trucks interact when they crash. Specifically, the authors are fixing a problem with how they simulate the "friction" between the two colliding streams of matter.

Here is the breakdown in simple terms:

1. The Setup: Three Streams of Traffic

In the old way of thinking, physicists modeled the collision as three separate "fluids" (like three different rivers flowing):

• The Projectile: The front half of the incoming truck (moving fast forward).
• The Target: The back half of the incoming truck (moving fast backward).
• The Fireball: The hot, messy pile of debris created right in the middle where they hit.

In a perfect world, these three rivers would stay separate. But in reality, they crash into each other, slow down, and mix. This interaction is called friction.

2. The Problem: The "All-or-Nothing" vs. "Keep Your Own" Models

The authors looked at two existing ways to calculate this friction, and both had flaws:

• The "All-or-Nothing" Model (Csernai): Imagine that every time a particle from the front truck hits a particle from the back truck, everything (energy, momentum, and the truck's license plate) instantly gets dumped into the middle Fireball.

  • The Flaw: This makes the middle too crowded and stops the trucks too fast. It doesn't explain why some parts of the trucks (the "baryons" or heavy particles) seem to pass through the crash and keep moving, a phenomenon called baryon transparency.

• The "Keep Your Own" Model (IMS): Imagine that when the trucks crash, the heavy parts (baryons) stay attached to their original trucks, and only the light debris (pions) flies into the middle.

  • The Flaw: This keeps the trucks moving too well. It fails to explain why the middle gets hot enough to create the right amount of new particles. It's too rigid.

3. The New Solution: The "Charge Transfer" Friction

The authors introduced a new, smarter rule called Charge Transfer Friction.

Think of it like a traffic merge zone with a smart gatekeeper.

• When particles from the two trucks collide, the gatekeeper looks at their speed (momentum).
• If a particle is moving very fast (like a sports car), it stays with its original truck.
• If a particle is moving slowly (like a slow-moving truck), it gets "transferred" to the middle Fireball.

Why is this better?
It creates a more realistic picture. It allows some heavy particles to get stuck in the middle (creating a dense core) while others keep moving. This matches what we see in real experiments much better than the old "all-or-nothing" or "keep your own" rules.

4. The Missing Ingredient: Viscosity (The Honey Effect)

Even with this new friction rule, the simulation was still missing something. The computer predicted fewer particles in the middle than what experiments actually saw.

The authors realized they needed to add viscosity (thickness).

• Analogy: Imagine the collision soup is water. If you stir water, it flows easily. But if you stir honey, it resists the flow, creates heat, and generates more turbulence.
• In the simulation, adding shear viscosity (making the fluid act a bit like honey) caused the middle Fireball to expand sideways more than it did lengthwise. This trapped more energy in the middle, creating more particles and matching the real-world data perfectly.

5. The Result: A Better Map of the Universe

By combining the new "smart gatekeeper" friction rules with the "honey-like" viscosity, the authors created a simulation that:

1. Correctly predicts how many particles are created in the middle.
2. Correctly predicts how the heavy particles slow down and spread out.
3. Works across different collision speeds (energies).

In a nutshell:
The paper is about fixing the "physics engine" for heavy-ion collisions. They replaced a clumsy, rigid rule for how colliding matter interacts with a flexible, speed-dependent rule, and then added a bit of "thickness" to the fluid to make the simulation match reality. This helps scientists understand the fundamental laws of matter under extreme conditions.

Technical Summary

1. Problem Statement

Heavy-ion collisions at intermediate energies ($\sqrt{s_{NN}} \sim 10 - 100$ GeV) present a unique challenge for theoretical modeling. Unlike high-energy collisions (e.g., at LHC) where distinct timescales allow for a clear separation between initial state, hydrodynamic evolution, and hadronic transport, intermediate-energy collisions involve simultaneous energy deposition, equilibration, and hydrodynamic evolution.

• Baryon Transparency: A key feature at these energies is "baryon transparency," where the colliding nuclei pass through each other, resulting in a double-peak structure in the net-baryon rapidity distribution.
• The Multi-Fluid Approach: To model this, the system is described as three interacting fluids: a projectile fluid, a target fluid, and a central "fireball" fluid.
• The Core Issue: The interaction between these fluids is modeled via "friction terms" (source terms in the hydrodynamic equations). Existing friction models have significant limitations:
  • Csernai Model: Assigns all scattered particles to the fireball. This fails to reproduce baryon transparency because it stops baryon charge as strongly as energy.
  • IMS (Ivanov-Mishustin-Satarov) Model: Keeps all baryon charge in the projectile/target fluids. While it reproduces the double-peak structure, it may decelerate the fluids too much and lacks flexibility in probing the Equation of State (EoS) at finite baryon density in the fireball.
  • Ideal Hydrodynamics: Previous multi-fluid simulations assumed ideal (non-viscous) fluids, which often led to discrepancies in particle multiplicity and flow observables.

2. Methodology

The authors developed and implemented a new "Charge Transfer" (CT) friction model within the MUFFIN multi-fluid solver (based on vHLLE), coupled with the SMASH hadronic cascade for the final stage.

A. The Charge Transfer Friction Model

The CT model generalizes previous approaches by allowing outgoing nucleons from projectile-target scatterings to be assigned to any of the three fluids based on their momentum (rapidity), rather than rigidly assigning them to the projectile/target or the fireball.

• Assignment Rules:
  • Outgoing nucleons with rapidity $|y| < \beta y_{in}$ are assigned to the fireball.
  • Outgoing nucleons with $|y| > \beta y_{in}$ remain in the projectile or target fluids.
  • Here, $y_{in}$ is the beam rapidity and $\beta$ is a free parameter ($0 \le \beta \le 1$).
• Energy Partitioning: A second parameter, $\alpha$, controls the fraction of energy lost by the nucleon (due to rapidity reduction) that is transferred to the fireball to create new particles, versus the energy retained as transverse momentum.
• Mathematical Formulation: The friction terms ($F^\nu$) and charge transfer terms ($R$) are derived from kinetic theory using specific cross-section integrals ($\bar{\sigma}_R, \bar{\sigma}_P, \bar{\sigma}_E$) over the rapidity ranges defined by $\beta$ and $\alpha$.

B. Dissipative Effects (Shear Viscosity)

Recognizing that ideal hydrodynamics underestimates entropy production, the authors incorporated shear viscosity into the internal evolution of the fluids.

• They used a parametrization of the specific shear viscosity $\tilde{\eta} = \eta/(\epsilon + p)$ that depends on the baryon chemical potential ($\mu_B$), based on Bayesian analysis from Ref. [31].
• The model includes the shear stress tensor in the evolution equations and uses Grad's 14-moment ansatz for non-equilibrium corrections ($\delta f$) at particlization.

C. Simulation Setup

• Equation of State: Chiral model EoS constrained to zero strangeness and a fixed electric charge-to-baryon ratio ($n_Q = 0.4 n_B$).
• Unification: A procedure is employed to merge fluids when their flow velocities become degenerate, preventing artificial increases in degrees of freedom.
• Validation: Simulations were performed for central (0-5%) Au+Au and Pb+Pb collisions at $\sqrt{s_{NN}} = 7.7, 19.6, 39, 62.4$ GeV.

3. Key Contributions

1. Novel Friction Model: Introduction of the Charge Transfer (CT) friction, which interpolates between the Csernai and IMS models via the parameter $\beta$. This allows for a more physically consistent description of fluids separated in momentum space.
2. Finite Baryon Density in Fireball: Unlike the IMS model, the CT model allows the fireball to acquire a finite net-baryon density, making it suitable for exploring the QCD Equation of State at finite $\mu_B$.
3. Inclusion of Dissipation: This work presents the first results of viscous multi-fluid dynamics. The authors demonstrate that including shear viscosity is crucial for resolving tensions between different observables.
4. Parameter Scans: A systematic exploration of the new parameters ($\alpha, \beta$) and scaling factors ($\xi_{pt}, \xi_f$) to understand their impact on baryon stopping and particle production.

4. Results

• Benchmarking at 19.6 GeV:
  • Csernai Model: Reproduces charged hadron multiplicity but fails to show the double-peak structure in net protons (excessive baryon stopping).
  • IMS Model: Reproduces the double-peak structure but underestimates charged hadron multiplicity at midrapidity.
  • CT Model (Ideal): With optimal parameters ($\alpha=0.7, \beta=0.1$), it reproduces the net-proton double peak but significantly underestimates the charged hadron multiplicity ($dN_{ch}/d\eta$).
• Role of Shear Viscosity:
  • Including shear viscosity significantly increases entropy production and charged hadron multiplicity, bringing the CT model results into excellent agreement with PHOBOS data.
  • Crucially, shear viscosity has a minimal impact on the net-proton distribution, preserving the double-peak structure. This resolves the previous tension where parameters fitting one observable ruined the other.
• Predictive Power:
  • The model successfully reproduces experimental data across a wide energy range (7.7 to 62.4 GeV) for both pseudorapidity distributions and net-proton rapidity distributions.
  • It correctly captures the transition from a single peak at low energy to a distinct double peak at higher energies.
  • Elliptic Flow ($v_2$): The inclusion of viscosity corrects the previous overestimation of elliptic flow, bringing predictions close to STAR data up to $p_T \approx 1$ GeV. However, at higher $p_T$, $v_2$ is suppressed too much, suggesting a need for bulk viscosity or improved shear parametrization.

5. Significance

This paper represents a significant advancement in the theoretical description of heavy-ion collisions at intermediate energies (relevant for the Beam Energy Scan program at RHIC and future experiments at FAIR/NICA).

• Physical Consistency: The CT friction model provides a more rigorous link between microscopic scattering rules and macroscopic fluid dynamics by respecting momentum-space separation.
• Equation of State Probing: By allowing baryon charge to accumulate in the fireball, the model enables more accurate constraints on the QCD Equation of State at high baryon density.
• Necessity of Dissipation: The study conclusively demonstrates that dissipative effects (shear viscosity) are essential in multi-fluid dynamics. Ignoring them leads to a fundamental trade-off where one cannot simultaneously describe particle production and baryon stopping.
• Future Directions: The work establishes a baseline for future refinements, including the incorporation of bulk viscosity, charge diffusion, and more sophisticated treatments of inter-fluid friction effects on dissipative quantities.