Machine learning the impact parameter in heavy-ion collisions at $\sqrt{s_{\rm NN}}$ = 4 and 11 GeV: a cross-check study with UrQMD, AMPT, and JAM

Technical Summary: Machine Learning the Impact Parameter in Heavy-Ion Collisions at $\sqrt{s_{NN}} = 4$ and 11 GeV

Problem Statement
Determining the collision centrality, defined by the impact parameter ($b$), is a fundamental prerequisite for analyzing heavy-ion collision (HIC) experiments, particularly at intermediate energies ($\sqrt{s_{NN}} \approx$ a few GeV) relevant to facilities like STAR-FXT, NICA/MPD, and FAIR/CBM. Unlike at ultra-relativistic energies, the classical Glauber model, which relies on the eikonal approximation, becomes unreliable at these lower energies due to the comparable de Broglie wavelength and nuclear size. Furthermore, traditional centrality determination methods often rely on fitting experimental observables (e.g., charged multiplicity) to transport model simulations. This approach is inherently model-dependent; different transport models (e.g., UrQMD, AMPT, JAM) can yield significantly different particle yields and distributions for the same impact parameter. Consequently, a centrality calibration derived from one model may not generalize to another or to real experimental data, introducing systematic uncertainties that can rival or exceed those from the nuclear equation of state.

Methodology
The authors employ a supervised and unsupervised machine learning (ML) framework to reconstruct the impact parameter in Au+Au collisions at $\sqrt{s_{NN}} = 4$ and 11 GeV.

• Data Generation: Labeled datasets were generated using three distinct transport models: the ultrarelativistic quantum molecular dynamics (UrQMD) model, the multiphase transport (AMPT) model, and the JAM model. To maximize the parameter space and test robustness, six specific configurations were utilized:
  • UrQMD with three setups: Mean-field + Hard-sphere (MH), Cascade + Hard-sphere (CH), and Cascade + Woods-Saxon (CW).
  • AMPT with the string-melting version.
  • JAM with two setups: Cascade mode with Woods-Saxon (C) and Mean-field mode with soft EoS (M).
  • For each configuration, 10,000 Au+Au events were simulated. The dataset was split into 6,000 training events and 4,000 test events.
• Input Features: Six event-by-event observables were selected as input features: the yields of charged pions ($N_{\pi^-}, N_{\pi^+}$), the mid-rapidity yields of charged pions ($n_{\pi^-}, n_{\pi^+}$), and the total transverse momenta of charged pions ($p_{t\pi^-}, p_{t\pi^+}$). These were chosen for their measurability and relatively lower event-by-event fluctuations compared to other observables.
• Algorithms:
  • Supervised Learning: The LightGBM (Light Gradient Boosting Machine) algorithm was used for regression (predicting continuous $b$) and classification (assigning events to centrality classes).
  • Unsupervised Learning: The K-means clustering algorithm was applied to partition events into centrality classes based solely on the input observables, without predefined $b$ labels.
• Cross-Validation Strategy: A rigorous cross-check was performed where models trained on data from one transport model (or specific setup) were tested on data generated by different models. This was compared against a traditional baseline method: a third-order polynomial fit relating charged pion yield to impact parameter.

Key Results

1. Regression Performance:

   • The LightGBM algorithm achieved a Mean Absolute Error (MAE) between 0.2 and 0.4 fm across all scenarios.
   • Crucially, this high accuracy persisted even in cross-model testing (e.g., training on UrQMD/CH and testing on AMPT). The MAE remained low, indicating strong generalization capabilities.
   • In contrast, the traditional polynomial fitting method performed well only when training and testing data came from the same model. When applied to data from different models, the polynomial fit failed significantly, producing large deviations because the fitted parameters were not transferable across models with different underlying physics assumptions.

2. Classification Performance:

   • The ML model successfully classified events into six centrality classes (0–5%, 5–10%, ..., >60%).
   • In cross-model validation, the predicted fractions of events in each centrality class deviated from the true simulation values by only ~1%.
   • The ML-predicted impact parameter distributions maintained a smooth, Gaussian-like shape consistent with the original simulations, demonstrating that the algorithm learned the general statistical patterns of HICs rather than memorizing specific model artifacts.

3. Unsupervised Clustering:

   • The K-means algorithm autonomously identified six distinct clusters in the feature space (normalized pion multiplicity vs. normalized transverse momentum) that corresponded directly to the six centrality classes.
   • These clusters showed minimal overlap and aligned with the physical expectation that central collisions (small $b$) produce high multiplicity and transverse momentum, while peripheral collisions (large $b$) produce low values.
   • This was achieved without any reliance on predefined model-based binning or ground-truth impact parameters during the clustering process.

Significance and Claims
The paper claims that machine learning offers a robust, model-independent solution for impact parameter determination in the intermediate energy regime. The primary significance lies in the demonstrated generalization ability: an ML algorithm trained on data from one transport model can accurately predict impact parameters for data generated by entirely different models. This addresses the critical issue of model dependence that plagues traditional centrality determination methods.

The authors assert that this approach:

• Surpasses the limitations of the Glauber model and traditional polynomial fitting at intermediate energies.
• Provides a data-driven framework that can be generalized to handle real experimental data from facilities like STAR-FXT, NICA, and FAIR.
• Establishes a foundation for more reliable extraction of nuclear matter properties (such as the equation of state) by reducing systematic uncertainties associated with centrality selection.

The study concludes that while the current work uses transport model data, the strong cross-model robustness suggests the method is ready for application to experimental observables, potentially improving the precision of future heavy-ion collision experiments.