Bayesian Analyses of Proton Multiple Flow Components in Intermediate Heavy Ion Collisions with Momentum-Dependent Interactions

Using a Bayesian framework with a Gaussian Process emulator to analyze Au+Au collision data at 1.23 GeV/nucleon, this study demonstrates that a momentum-dependent transport model constrained by proton flow observables favors a soft nuclear equation of state and mild in-medium suppression of baryon-baryon cross sections, while highlighting that momentum-independent models require significantly different parameters to reproduce the same data.

The Gist

The Big Picture: Smashing Atoms to Understand the Universe's "Glue"

Imagine you want to understand how a car engine works, but you can't take it apart. Instead, you crash two cars together at high speed and watch how the pieces fly apart. By studying the debris, you can figure out how strong the bolts were, how stiff the metal was, and how the engine reacted to the impact.

This is exactly what physicists do with Heavy Ion Collisions. They smash gold atoms (nuclei) together at nearly the speed of light. Inside these tiny, super-hot explosions, matter behaves in ways we've never seen before—like a super-dense fluid. The goal of this paper is to figure out the "rules of the road" for this dense matter.

The Two Main Characters: The "Stiffness" and the "Friction"

To simulate these crashes on a computer, scientists need to program two main rules into their model:

1. The Stiffness (Incompressibility, $K_0$): Imagine the nuclear matter as a giant spring.

   • Stiff Spring: It's very hard to squish. When you push it, it pushes back hard.
   • Soft Spring: It squishes easily and doesn't push back as hard.
   • Why it matters: This tells us about the "Equation of State" (EoS)—basically, how matter behaves under extreme pressure. This is crucial for understanding neutron stars (which are basically giant atomic nuclei).

2. The Friction (Scattering Cross-section, $X$): Imagine the particles inside the collision as people in a crowded dance hall.

   • High Friction ($X > 1$): The dancers bump into each other constantly, slowing down and changing direction easily.
   • Low Friction ($X < 1$): The dancers glide past each other easily, rarely bumping.
   • Why it matters: This determines how often particles collide inside the explosion, which changes how the debris flies out.

The Problem: The "Blind Men and the Elephant"

For a long time, scientists tried to figure out these two rules separately. They would say, "Let's assume the spring is stiff and see what happens," or "Let's assume the friction is high and see what happens."

But here's the catch: Stiffness and Friction are best friends.

• If the spring is stiff, it pushes particles out fast.
• If the friction is high, it also pushes particles out fast (because they bounce off each other more).

If you only look at the final result (how fast the particles fly), you can't tell if it's because the spring was stiff or because the friction was high. It's like trying to guess if a car went fast because the engine was powerful or because the road was slippery. You need a better way to solve this puzzle.

The Solution: A "Smart Guessing Machine" (Bayesian Analysis)

The authors of this paper used a powerful statistical tool called Bayesian Analysis. Think of this as a super-smart detective who doesn't just guess one answer, but calculates the probability of thousands of different scenarios.

1. The Simulator (The IBUU Model): They used a complex computer program that simulates the gold atom crash.
2. The "Speed Booster" (Gaussian Process Emulator): Running the full simulation takes forever. So, they built a "shortcut" (an emulator) that learns the patterns of the simulation and can predict the results instantly, like a weather app that predicts rain without running a full physics model.
3. The Reality Check (HADES Data): They compared their millions of computer simulations against real data from the HADES experiment (a real detector in Germany that actually smashed gold atoms together).

The "Flow" of the Debris

When the atoms smash, the particles don't just fly in a straight line; they swirl and flow like water. The scientists measured four different types of this "flow":

• Directed Flow ($F_1$): Like water rushing to one side.
• Elliptic Flow ($v_2$): Like water squeezing into an oval shape.
• Triangular Flow ($F_3$) & Quadrupole Flow ($v_4$): More complex, wobbly patterns.

By checking how well their computer models matched these specific swirls in the real data, they could narrow down the answers.

The Results: What Did They Find?

After running their "detective work," they found some surprising things:

1. The Spring is Soft: The data suggests the nuclear matter is not a super-stiff spring. It's relatively "soft" (a low value for $K_0$). This means the matter squishes easily under pressure.
2. The Friction is Normal (Slightly Low): The particles don't bump into each other more than they do in empty space. In fact, the collisions are slightly suppressed (the friction factor $X$ is around 0.9 to 1.0). The particles glide through the medium almost as if they are in a vacuum.
3. The "Momentum" Trick: The most important discovery is about how they modeled the forces.
   • When they used a simple model where forces don't change with speed (momentum-independent), they had to assume the spring was very stiff and the friction was very high to match the data.
   • But when they used a realistic model where forces change depending on how fast the particles are moving (momentum-dependent), they found the "soft spring" and "normal friction" scenario worked perfectly.

The Takeaway: Why This Matters

Think of the "momentum-dependent force" like a smart suspension system in a car.

• If you ignore the smart suspension (use a simple model), you have to assume the car is made of solid steel (stiff) and the tires are sticky (high friction) to explain why it handles turns the way it does.
• But if you include the smart suspension (momentum dependence), you realize the car is actually made of lighter material (soft) and the tires are smoother, and the suspension does the heavy lifting.

In simple terms: This paper proves that to understand the universe's densest matter, we must account for how fast particles are moving when they interact. If we ignore that speed factor, we get the wrong answers about how "stiff" the universe is.

This helps us understand neutron stars better. Since neutron stars are made of this same dense matter, knowing it is "soft" helps us predict how big they can get before collapsing into black holes.

Technical Summary

1. Problem Statement

Transport models used to simulate heavy-ion collisions (HICs) rely on two fundamental, yet tightly coupled, inputs:

1. The Single-Particle Mean Field: Related to the Nuclear Equation of State (EoS), specifically characterized by the incompressibility ($K_0$) at saturation density.
2. In-Medium Baryon-Baryon Scattering Cross Sections: Modified by the nuclear environment (Pauli blocking, effective mass changes).

The Challenge:

• Parameter Degeneracy: Traditional approaches often vary the EoS or the cross-section independently. However, these two inputs are intrinsically coupled in their impact on observables (like collective flow). Varying one while fixing the other leads to ambiguous or biased constraints.
• Momentum Dependence: Many models use momentum-independent mean fields, which may require artificially stiff EoS or large cross-section corrections to reproduce experimental data. The specific role of momentum dependence in the mean field and its interplay with in-medium scattering remains to be fully quantified.
• Data Utilization: Previous studies often focused on limited observables. There is a need to simultaneously constrain parameters using a comprehensive set of flow observables (directed, elliptic, triangular, and quadrupole flows) within a rigorous statistical framework.

2. Methodology

The authors employed a Bayesian statistical framework combined with a Gaussian Process (GP) emulator to analyze Au + Au collision data at a beam energy of 1.23 GeV/nucleon.

• Transport Model: The study utilized the Isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model.

  • Mean Field: Implemented a momentum-dependent mean-field potential (ImMDI) based on Hartree-Fock calculations with Gogny interactions.
  • Cross Sections: The in-medium baryon-baryon scattering cross sections were scaled by a modification factor $X$ ($\sigma_{medium} = X \cdot \sigma_{free}$).
  • Calibration: Model parameters were calibrated to empirical nuclear matter properties at saturation density ($\rho_0 = 0.16 \text{ fm}^{-3}$), including binding energy, symmetry energy, and effective mass.

• Bayesian Inference Setup:

  • Parameters: The incompressibility ($K_0$) and the cross-section modification factor ($X$) were treated as free parameters.
  • Priors: Uniform priors were used: $K_0 \in [100, 380]$ MeV and $X \in [0.5, 2.0]$.
  • Observables: The analysis compared model predictions against HADES Collaboration experimental data for proton collective flow in two centrality classes (10–20% and 20–30%). The observables included:
    • Slope of directed flow ($F_1$) and triangular flow ($F_3$) at mid-rapidity.
    • Elliptic flow ($v_2$) and quadrupole flow ($v_4$) at specific transverse momentum ($0.6 \le p_t \le 0.9$ GeV/c).
  • Emulator: Due to the computational cost of running IBUU simulations during Markov Chain Monte Carlo (MCMC) sampling, a Gaussian Process (GP) emulator was trained on 240 parameter sets (combinations of $X$ and $K_0$) to predict flow observables. The emulator showed high accuracy (Pearson correlation coefficients $>0.94$).

• Likelihood Function: Constructed using a Gaussian distribution comparing experimental data with emulator predictions, accounting for experimental errors, model statistical errors, and emulator uncertainties.

3. Key Contributions

• Simultaneous Constraint: Successfully performed a joint Bayesian analysis to simultaneously constrain the nuclear incompressibility ($K_0$) and the in-medium scattering modification factor ($X$), resolving the degeneracy often found in single-parameter studies.
• Momentum Dependence Impact: Provided a direct quantitative comparison between models with momentum-dependent (MDI) and momentum-independent mean fields, demonstrating how the choice of mean field alters the inferred EoS and cross-section properties.
• Multi-Observable Analysis: Systematically analyzed the sensitivity of higher-order flow components ($F_3, v_4$) alongside standard flows ($F_1, v_2$), revealing that higher-order flows provide tighter constraints on the cross-section modification factor.

4. Key Results

A. Constraints on Parameters (with Momentum-Dependent Mean Field)

• Incompressibility ($K_0$): The posterior distribution favors relatively soft nuclear EoS values. The most probable range is $K_0 < 210$ MeV, with a strong accumulation below 250 MeV.
• Cross-Section Modification ($X$): The inferred average values for $X$ fall in the range 0.9 – 1.0. This suggests a mild suppression of baryon-baryon cross sections in the medium at 1.23 GeV/nucleon.
  • Higher-order flows ($v_4$) prefer slightly lower $X$ values compared to $F_1$.
  • The results indicate that the in-medium suppression at this energy is weaker than predicted by some zero-temperature microscopic calculations (which often predict stronger suppression).

B. The Role of Momentum Dependence

• Comparison: When the analysis was repeated using a momentum-independent mean field:
  • The required $K_0$ shifted to stiffer values ($> 240$ MeV).
  • The required $X$ shifted to larger values ($\sim 1.3 – 1.6$).
• Interpretation: Momentum dependence in the mean field naturally enhances nuclear stopping power and collective flow. Therefore, models with momentum dependence do not need to invoke a stiff EoS or large cross-section scaling to reproduce the HADES data. Models without momentum dependence must artificially compensate with stiffer EoS and stronger cross-section corrections.

C. Correlations

• An anti-correlation was observed between $K_0$ and $X$. Both a stiffer EoS (higher $K_0$) and a larger cross section (higher $X$) increase nuclear stopping power. The Bayesian analysis successfully broke this degeneracy, identifying the "soft EoS / mild suppression" scenario as the most probable.

5. Significance

• Refining the Nuclear EoS: The study provides robust evidence that the nuclear EoS at intermediate densities is likely softer ($K_0 \approx 160–200$ MeV) when momentum-dependent interactions are correctly accounted for. This has implications for understanding neutron star structure and the core-collapse supernova mechanism.
• Medium Effects: The finding that $X \approx 1.0$ at 1.23 GeV suggests that medium modifications to the scattering cross section are less severe at intermediate energies than previously thought, possibly due to energy-dependent effects or the inclusion of inelastic channels.
• Methodological Advancement: The work demonstrates the critical necessity of using momentum-dependent mean fields in transport simulations to avoid biased extraction of nuclear matter properties. It also validates the use of GP emulators for efficient, high-dimensional Bayesian inference in nuclear physics.
• Future Directions: The authors highlight the need to extend these analyses to broader beam energies (0.1–3 GeV/nucleon) and include isospin-sensitive probes (e.g., neutron-to-proton flow ratios) to further constrain the symmetry energy and density dependence of the EoS.