Transport-based initial conditions for heavy-ion collisions at finite densities

This paper presents a unified framework within the X-SCAPE code that utilizes the SMASH transport model to generate event-by-event initial conditions for hydrodynamic simulations of heavy-ion collisions at finite densities, incorporating a 4D lattice-QCD equation of state and generalized out-of-equilibrium corrections to study conserved charge fluctuations in the Beam Energy Scan program.

The Gist

Imagine trying to understand what happens when two cars crash at high speed. In the world of physics, instead of cars, we smash together huge atomic nuclei (like gold atoms) to create a tiny, super-hot drop of liquid called Quark-Gluon Plasma (QGP). This liquid is so hot and dense that it behaves like the universe did just microseconds after the Big Bang.

For decades, physicists have been good at simulating this crash when the nuclei are moving near the speed of light. But what happens when they collide at lower speeds? The rules change, and the old "blueprints" for how the crash starts no longer work well.

This paper introduces a new, more realistic way to build the starting blueprint for these collisions, specifically for scenarios where there is a lot of "stuff" (matter) packed into the collision zone.

Here is the breakdown of their new approach using simple analogies:

1. The Old Way vs. The New Way

• The Old Way (Static Maps): Previously, scientists used simplified models to guess what the collision looked like at the very first split-second. It was like trying to predict a traffic jam by looking at a static map of the roads. It worked for high-speed crashes, but it missed the messy details of how cars (particles) actually stop, turn, and interact when things get crowded.
• The New Way (The Traffic Simulator): This paper uses a sophisticated computer program called SMASH. Think of SMASH as a high-definition video game engine that simulates every single particle in the nucleus. Instead of guessing the starting point, they let the particles crash, bounce, and interact in a "pre-game" phase. Only after they have danced around a bit do they hand the data over to the main simulation. This gives a much more accurate picture of the "initial conditions."

2. The Three Types of "Charge" (The Ingredients)

In these collisions, three things must be conserved (kept track of):

1. Baryon Number (B): Think of this as the "heavy stuff" (protons and neutrons).
2. Electric Charge (Q): The positive and negative electricity.
3. Strangeness (S): A weird property of certain heavy particles.

The Big Discovery:
The authors found that the "heavy stuff" (Baryons) behaves very differently from the "electricity" (Charge) and "strangeness."

• The Heavy Stuff: It's like a heavy freight train. It's hard to stop, so it tends to pile up in the middle. It doesn't fluctuate much; it's mostly about where the train stopped.
• The Electricity and Strangeness: These are like a swarm of bees or a bag of mixed jellybeans. Because it takes very little energy to create a pair of positive/negative particles (or strange particles), they pop in and out of existence constantly.
• The Result: The new simulation shows that while the "heavy stuff" is smooth, the "electricity" and "strangeness" are chaotic and bumpy. This creates a very "spotty" starting point for the fluid, which changes how the whole system evolves.

3. The "Stretchy" Smearing (The Lens)

When the simulation moves from the "particle game" (SMASH) to the "fluid physics" (Hydrodynamics), it has to turn individual particles into a smooth liquid.

• The Problem: Some particles are moving incredibly fast. In physics, fast-moving things look "squished" (Lorentz contraction) to an observer.
• The Fix: The team developed a new mathematical "lens" (called a Covariant Smearing Kernel) that accounts for this squishing.
• The Analogy: Imagine taking a photo of a fast-moving car. If you use a standard camera, the car looks blurry or stretched. If you use a special "speed camera" (the new kernel), you get a sharp, accurate picture of the car's shape, even though it's moving fast. This ensures the fluid simulation starts with the correct pressure and density.

4. The "Afterburner" (The Cleanup Crew)

Once the fluid cools down and turns back into individual particles (a process called "particlization"), the simulation doesn't stop.

• The Process: The particles are fed back into the SMASH simulator for a second round.
• The Analogy: Think of the fluid phase as the main explosion of a firework. The "afterburner" is the time after the explosion when the sparks are still flying, colliding, and fading away. This final stage is crucial because it determines what the detectors actually see on the ground.

Why Does This Matter?

This work is a major step toward mapping the Phase Diagram of Matter.

• Scientists are trying to find a "critical point" in the universe where matter changes phases (like water turning to ice, but for nuclear matter).
• To find this, they need to run experiments at different energies (speeds).
• This new method provides the most accurate "starting line" yet for these lower-energy experiments (like those at the RHIC Beam Energy Scan or future facilities in Germany).

In Summary:
The authors built a better "pre-game" simulator (SMASH) to generate realistic starting conditions for heavy-ion collisions. They discovered that at lower energies, the "electric" and "strange" parts of the collision are much more chaotic than the "heavy" parts. By using a new mathematical lens to handle fast-moving particles, they ensure that the fluid simulation starts with the right physics, helping scientists better understand the fundamental building blocks of our universe.

Technical Summary

1. Problem Statement

Relativistic heavy-ion collision experiments, particularly those in the Beam Energy Scan (BES) program at RHIC and future facilities like FAIR, aim to map the Quantum Chromodynamics (QCD) phase diagram at finite net baryon densities. Traditional initial condition models for hydrodynamic simulations often rely on:

• Eikonal approximations and parton saturation physics (e.g., IP-Glasma), which lose predictive power at lower collision energies where saturation scales become non-perturbative.
• Glauber-type models, which lack descriptions of complex baryon transport, longitudinal fluctuations, and the detailed stopping of individual nucleons.
• Simplified charge conservation, often treating only net baryon number or neglecting the independent evolution of net electric charge ($Q$) and net strangeness ($S$).

There is a critical need for a microscopic, dynamical framework that can generate event-by-event initial conditions for the energy-momentum tensor and three conserved charge currents ($B, Q, S$) at finite densities, bridging the gap between non-equilibrium transport and relativistic hydrodynamics.

2. Methodology

The authors developed a unified hybrid framework integrated within the X-SCAPE code, orchestrating a four-stage workflow: SMASH (Pre-equilibrium) $\to$ MUSIC (Hydrodynamics) $\to$ iSS (Particlization) $\to$ SMASH (Afterburner).

A. Initial Conditions: SMASH Transport Model

• Model: The microscopic hadronic transport model SMASH (version 3.1) is used to simulate the non-equilibrium evolution of the system from the initial nuclear overlap.
• Physics: It includes all hadronic degrees of freedom up to 2.35 GeV, string fragmentation (via Pythia 8) for high-energy interactions, and resonance decays for lower energies.
• Initialization: Nuclei are sampled from Woods-Saxon distributions. The system evolves until a constant proper time $\tau_0$ (determined by nuclear passing time), where particles are extracted to initialize hydrodynamics.
• Fluctuations: The model naturally captures baryon stopping, longitudinal fluctuations, and the development of pressure anisotropies.

B. Hydrodynamic Evolution

• Solver: The (3+1)D relativistic viscous hydrodynamic code MUSIC solves the conservation equations for energy-momentum ($T^{\mu\nu}$) and three conserved currents ($J^\mu_B, J^\mu_Q, J^\mu_S$).
• Equation of State (EoS): A 4D lattice-QCD-based EoS (NEOS-4D) is employed, where pressure $P$ depends on energy density ($e$) and chemical potentials for baryon, electric charge, and strangeness ($\mu_B, \mu_Q, \mu_S$). This allows for the independent propagation of all three charge densities.
• Smearing Kernels: To convert discrete hadrons from SMASH into continuous hydrodynamic source terms, the authors implemented a covariant smearing kernel. Unlike standard Gaussian kernels, this kernel accounts for Lorentz contraction in the direction of the hadron's motion, providing a more physically accurate density profile for fast-moving particles.
• Viscosity: Shear and bulk viscosities are evolved using DNMR theory, with parameters tuned to lattice QCD constraints.

C. Particlization (Freeze-out)

• Procedure: The Cooper-Frye procedure is used to convert fluid cells into hadrons when the energy density drops below a switching threshold ($e_{sw} = 0.35$ GeV/fm$^3$).
• Out-of-Equilibrium Corrections ($\delta f$): A major theoretical advancement is the generalization of $\delta f$ corrections to finite densities with three conserved charges.
  • The authors derived $\delta f$ using both Grad's 14-moment method and the Chapman-Enskog (CE) expansion.
  • These derivations include independent coefficients for shear stress and bulk pressure, as well as terms coupling to the gradients of $\mu_B, \mu_Q,$ and $\mu_S$, ensuring the conservation of all macroscopic quantities during the transition.

D. Afterburner

• Produced hadrons are fed back into SMASH for the dilute, non-equilibrium "afterburner" phase, simulating rescatterings and decays until kinetic freeze-out.

3. Key Contributions

1. Unified 4D Framework: Established the first fully integrated framework (X-SCAPE) capable of handling event-by-event fluctuations of three conserved charges ($B, Q, S$) from transport initialization through hydrodynamic evolution to final-state observables.
2. Generalized $\delta f$ Corrections: Derived and implemented out-of-equilibrium corrections for particlization that explicitly account for finite baryon, electric, and strangeness densities, ensuring strict conservation laws are met during the fluid-to-particle transition.
3. Covariant Smearing Kernel: Introduced a covariant smearing kernel that incorporates Lorentz contraction for source terms, improving the physical description of initial state gradients compared to standard Gaussian smearing.
4. Transport-Based Initial Conditions: Demonstrated that hadronic transport models (SMASH) provide a unique and necessary description of initial charge fluctuations, particularly for $Q$ and $S$, which are driven by local pair production rather than just initial stopping.

4. Key Results

• Charge Fluctuations: The SMASH initial conditions reveal that net electric charge ($Q$) and net strangeness ($S$) exhibit significantly larger local fluctuations compared to net baryon density ($B$). This is because producing charged meson pairs ($\pi^+\pi^-$) is energetically cheaper than baryon-antibaryon pairs, leading to a "fluctuation-dominated" distribution for $Q$ and $S$, whereas $B$ is dominated by initial-state stopping.
• Hydrodynamic Evolution:
  • At $\sqrt{s_{NN}} = 200$ GeV, the system shows flux-tube-like structures and rapid expansion.
  • At lower energies ($\sqrt{s_{NN}} = 19.6$ and $7.7$ GeV), the fireball size remains roughly constant due to a balance between radial flow development and cooling. The system probes a wide range of the QCD phase diagram in the $n_Q$ and $n_S$ directions.
• Impact of Smearing Kernels:
  • The covariant kernel produces sharper initial gradients, leading to stronger radial flow and flatter transverse momentum ($p_T$) spectra for identified particles (pions and protons) compared to the Gaussian kernel.
  • However, the final-state pseudorapidity distributions of conserved charges are similar between the two kernels, as both account for longitudinal Lorentz contraction.
• Impact of $\delta f$ Corrections:
  • Grad's moment method yields larger corrections to particle spectra at high $p_T$ (quadratic growth) compared to the Chapman-Enskog expansion.
  • Including $\delta f$ corrections generally suppresses the elliptic flow ($v_2$) of charged hadrons.
  • The CE correction tends to flatten spectra (shear dominance), while Grad's correction steepens them (bulk dominance).

5. Significance

This work represents a significant step forward in the theoretical modeling of heavy-ion collisions at finite baryon density. By moving beyond simplified initial conditions and single-charge conservation, the authors provide a robust tool for:

• Precision Phenomenology: Enabling systematic comparisons with BES data to constrain the QCD Equation of State and transport coefficients (viscosity, diffusion) in the high-density region.
• Exploring the Phase Diagram: The ability to track $B, Q,$ and $S$ independently allows for a more accurate mapping of the QCD phase diagram, including the search for the critical point and the study of the "core-corona" dynamics.
• Future Experiments: The framework is specifically designed to support upcoming experiments at FAIR and NICA, where finite density effects are dominant, and traditional high-energy approximations fail.

The integration of these developments into the open-source X-SCAPE framework ensures reproducibility and facilitates future studies of core-corona dynamics and concurrent hydrodynamic/transport descriptions.