Effects of the centrality determination method for the equation of state and nucleonic observables from Au+Au collisions at $\sqrt{s_{NN}}$ = 2.4 GeV

This study utilizes UrQMD simulations of Au+Au collisions at $\sqrt{s_{NN}}$ = 2.4 GeV to demonstrate that the choice of centrality determination method significantly impacts the extraction of the nuclear equation of state, revealing that Glauber Monte Carlo-based approaches introduce larger uncertainties than the equation of state itself, whereas geometrical interpretations remain consistent with dynamical multiplicity selection.

The Gist

Imagine you are trying to understand the recipe for a very dense, hot soup (nuclear matter) by watching two giant bowls smash into each other. Physicists call this a "heavy-ion collision." When these bowls (gold atoms) crash, they squash the matter inside, creating conditions found only in the cores of neutron stars or the early universe.

The goal of this paper is to figure out how "stiff" or "squishy" this nuclear soup is. This property is called the Equation of State (EoS). Think of it like asking: Is the soup made of jelly (soft) or concrete (hard)?

However, there's a big problem: To measure the soup's properties accurately, you need to know exactly how hard the two bowls hit each other. Did they graze each other (a "peripheral" collision) or smash dead center (a "central" collision)?

The Problem: How Do We Measure the "Hit"?

In a real experiment, you can't see the center of the collision directly. It happens too fast and is too small. Instead, scientists have to guess how hard the hit was by counting the "debris" (particles) flying out.

The authors of this paper asked a simple question: "Does it matter how we guess the hardness of the hit? Will our guess change our conclusion about whether the soup is jelly or concrete?"

They tested three different ways to guess the "hardness" (centrality) of the collision:

1. The Debris Count (Mch): "We'll assume the more particles we see, the harder the hit was."
2. The Geometric Guess (bf): "We'll assume the hit was this hard based on a simple drawing of how the balls overlap."
3. The Computer Model Guess (br): "We'll use a famous computer simulation (Glauber model) to guess the hit based on how many people (particles) were involved."

The Experiment: A Virtual Crash Test

The researchers didn't smash real atoms. They used a super-computer simulation called UrQMD to crash gold atoms together at a specific energy (2.4 GeV). They ran the crash twice:

• Once with "Jelly Soup" (Soft EoS).
• Once with "Concrete Soup" (Hard EoS).

Then, they looked at the results using the three different guessing methods mentioned above.

The Findings: The "Guessing Game" Matters

Here is what they discovered, using simple analogies:

1. The "Debris Count" vs. The "Geometric Guess" (Mch vs. bf)
When they used the Debris Count and compared it to the simple Geometric Guess, the results were surprisingly similar.

• Analogy: Imagine trying to guess how hard a car crash was by counting broken headlights. If you count the headlights, you get a pretty good idea of the crash severity, and it matches up well with a simple drawing of the car overlap.
• Result: The uncertainty in the "guess" didn't mess up the measurement of the soup's stiffness. The "Jelly" and "Concrete" results stayed distinct.

2. The "Debris Count" vs. The "Computer Model" (Mch vs. br)
This is where things got messy. When they used the Debris Count and compared it to the Computer Model (Glauber), the results were very different.

• Analogy: Imagine using a high-tech weather app to guess the crash severity. The app assumes that "more broken parts = harder crash" in a way that works for big highway crashes, but fails for small, low-speed fender benders.
• Result: At this specific energy level (2.4 GeV), the computer model was wrong. It misidentified how hard the collisions were. Because of this bad guess, the difference between "Jelly Soup" and "Concrete Soup" became blurred. In fact, the error caused by using the wrong guessing method was bigger than the actual difference between the jelly and the concrete!

The Main Takeaway

The paper concludes that if you want to measure the "stiffness" of nuclear matter accurately:

• Don't trust the old computer model (Glauber) for these specific types of collisions. It's like using a map of a city to navigate a forest; the rules don't apply.
• Stick to the geometric logic or the direct debris count. These methods align better with reality at these energy levels.

In short: If you use the wrong ruler to measure the crash, you might think the soup is "Jelly" when it's actually "Concrete," or vice versa. The way you define the collision is just as important as the physics you are trying to measure. The authors warn that to get the recipe right, we need a consistent way to match the "debris count" to the "actual crash geometry," especially at lower energies.

Technical Summary

Technical Summary: Effects of Centrality Determination on Nucleonic Observables in Au+Au Collisions at $\sqrt{s_{NN}} = 2.4$ GeV

Problem Statement
In intermediate-energy heavy-ion collision (HIC) analyses, the determination of collision centrality remains a primary source of systematic uncertainty. This is particularly critical when probing the nuclear equation of state (EoS) at supra-saturation densities. While theoretical simulations utilize the impact parameter ($b$) as a direct input, experimental centrality is inferred from final-state observables (e.g., charged-particle multiplicity, $M_{ch}$) using theoretical models. There is no strictly one-to-one mapping between impact parameter and these observables. Previous work has indicated that the true impact parameter distribution of selected events possesses a width that significantly influences collective flow. Furthermore, the applicability of standard centrality determination methods, such as the Glauber Monte Carlo (MC) model, at low energies (e.g., 2.4 GeV) is questionable, as the model assumes particle multiplicity is proportional to the number of participants—an assumption more valid at higher energies. This study aims to quantitatively assess the uncertainties associated with different centrality determination methods and evaluate their impact on final-state EoS-sensitive observables.

Methodology
The study utilizes the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model to simulate Au+Au collisions at $\sqrt{s_{NN}} = 2.4$ GeV (corresponding to a beam energy of 1.23A GeV).

• Equation of State (EoS): Two momentum-dependent EoS scenarios are employed: a soft EoS ($K_0 = 200$ MeV, labeled SM) and a hard EoS ($K_0 = 380$ MeV, labeled HM).
• Centrality Determination Methods: Three distinct methods are compared to select event samples within specific centrality classes (0–10%, 10–20%, 20–30%, and 30–40%):
  1. $M_{ch}$: Selection based on the multiplicity of all charged particles.
  2. $b_f$: A geometrical interpretation based on the input impact parameter range.
  3. $b_r$: A method derived from the Glauber MC model (based on Ref. [7]), which maps multiplicity to impact parameter ranges.
• Observables: The study analyzes the yield and rapidity distributions of free protons, directed flow ($v_1$) and elliptic flow ($v_2$) coefficients (specifically the slope $v_{11}$ and mid-rapidity value $v_{20}$), transverse momentum ($p_T$) distributions, and mean transverse momentum ($\langle p_T \rangle$).

Key Results

• Impact Parameter Distributions: Significant discrepancies exist between the real impact parameter distributions of event samples selected by $M_{ch}$, $b_f$, and $b_r$. While $M_{ch}$ and $b_f$ show relatively consistent distributions, the deviation between $M_{ch}$ and $b_r$ is substantial. Specifically, for peripheral collisions (e.g., 30–40%), the fraction of events in the $b_r$ sample with impact parameters outside the expected range differs by up to 60% compared to $M_{ch}$ samples.
• Proton Yields: The free proton yield is sensitive to both the EoS stiffness and the centrality method.
  • When using the geometrical method ($b_f$), the uncertainty introduced by centrality selection is weaker than the effect induced by the EoS.
  • When using the Glauber MC-based method ($b_r$), the influence of centrality-related uncertainties becomes more pronounced than the EoS effect, particularly in peripheral collisions. The ratio of yields between $M_{ch}$ and $b_r$ selections increases significantly with centrality.
• Collective Flow: The directed flow slope ($v_{11}$) and elliptic flow ($v_{20}$) are primarily governed by the stiffness of the EoS. While $v_{11}$ is largely insensitive to the centrality determination method, $v_{20}$ shows a slight sensitivity, with $b_r$-selected events yielding slightly larger values than $M_{ch}$-selected events.
• Transverse Momentum: The $p_T$ distribution and mean transverse momentum follow trends similar to the yield. The $p_T$ distribution is strongly dependent on the centrality definition when using $b_r$, whereas the mean transverse momentum $\langle p_T \rangle$ is found to be relatively insensitive to both the EoS and the centrality determination method.

Significance and Conclusions
The paper concludes that a rigorous and consistent mapping between charged-particle multiplicity ($M_{ch}$) and impact parameter is essential for imposing quantitative constraints on the high-density nuclear EoS.

• Validity of Methods: At SIS18 energies ($\sqrt{s_{NN}} = 2.4$ GeV), the geometrical interpretation of centrality ($b_f$) remains valid and consistent with dynamical multiplicity selection. In contrast, the Glauber MC-based centrality determination ($b_r$) becomes unreliable in this energy regime, likely due to the breakdown of the assumption that particle multiplicity is strictly proportional to the number of participants.
• Implications for EoS Constraints: The study demonstrates that using an inappropriate centrality determination method (specifically Glauber MC at low energies) can introduce systematic uncertainties that rival or exceed the physical effects of the EoS itself. Therefore, future constraints on the nuclear EoS must account for the specific centrality selection method used, ensuring that the mapping between experimental observables and geometric centrality is physically justified for the energy regime under investigation.

Future Outlook
The authors suggest that future work should employ Bayesian inference techniques to quantitatively assess systematic deviations between simulations and experimental data, with a focus on fluctuations and correlations across different centrality classes. Additionally, a more detailed investigation into the energy dependence and model assumptions underlying Glauber MC-based methods is necessary to improve the reliability of EoS studies at low and intermediate energies.