Resolving Ratio Redundancy in Chemical Freeze-out Studies with Principal Component Analysis and Bayesian Calibration

TL;DR

The paper tackles the problem of ratio-induced ambiguities in extracting chemical freeze-out parameters from hadron yields in heavy-ion collisions. It develops a PCA–Bayesian framework that decorrelates the ratio space via log-ratios and principal components, and uses a Gaussian Process emulator to efficiently calibrate an HRG model across a four-parameter space $(T,\mu_B,\mu_S,\mu_Q)$ while incorporating Sobol sensitivity to identify the most informative observables. The results demonstrate an energy-dependent transition from chemical-potential–dominated to temperature-controlled freeze-out, with $T$ becoming the dominant parameter at high energies and the posteriors for the freeze-out parameters aligning with previous HRG analyses. The framework provides a statistically rigorous, information-preserving, and scalable approach to extract freeze-out conditions, and it can be extended to centrality dependence and interacting HRG formulations, offering a robust tool for QCD phase-structure studies in heavy-ion collisions.$

Abstract

We introduce a Principal Component Analysis (PCA)--Bayesian framework for extracting chemical freeze-out conditions in relativistic heavy-ion collisions that resolves long-standing ambiguities in hadron-ratio--based analyses. By constructing all possible hadron-yield ratios from a chosen set of species and transforming them into an orthogonal PCA basis, the method removes linear redundancies and eliminates the information loss and systematic uncertainties associated with ratio selection. Energy-wise Bayesian calibration of the Hadron Resonance Gas (HRG) model is then performed directly in this decorrelated space, with a Gaussian Process emulator enabling fast and accurate model evaluations. A detailed Sobol sensitivity analysis, together with the PCA loading structure, identifies the most informative ratio combinations and reveals a transition from chemical-potential--dominated to temperature-controlled freeze-out with increasing $\sqrt{s_{NN}}$. The calibrated model reproduces all measured ratios, and the extracted freeze-out parameters are consistent with previous HRG determinations.

Figures (10)

• Figure 1: PCA diagnostics for the hadron yield ratio ensemble. Top: Cumulative variance explained as a function of the number of principal components, with dashed lines marking the 96% and 99% thresholds. Middle: Comparison between the standardized log-ratio data and the reconstruction from the retained components. Bottom: Ratio of the reconstructed to the original data.
• Figure 2: Loading matrix representing the first ten principal components (PC1--PC10) for identified-particle yield log-ratios at $\sqrt{s_{NN}}=39$ GeV. The eigenvalues $\lambda_i$ are indicated for each component. The color bar shows the contribution of each log-ratio to each principal component.
• Figure 3: GP Emulator diagnostics. Top-left: HRG model calculations versus emulator predictions for various hadron species , with error bars scaled to the predicted uncertainty. The dashed line denotes perfect agreement. Bottom-left: Ratio of model to emulator predictions, showing deviations tightly clustered around unity. Right: Distribution of normalized residuals with a standard normal overlay, demonstrating statistically consistent uncertainty estimates.
• Figure 4: Energy-wise sensitivities of the chemical freeze-out parameters obtained from Sobol total indices ($S_T$). Each column shows the nine algebraically independent hadron yield ratios with the largest $S_T$ values for the corresponding freeze-out parameter. The common color bar indicates the magnitude of $S_T$.
• Figure 5: Posterior distributions from the Bayesian calibration at $\sqrt{s_{NN}}=39$ GeV (Case I). Diagonal: one-dimensional marginalized posteriors with mean (dashed) and median (solid) lines. Off-diagonal: two-dimensional joint posteriors with 68 Inset: evolution of $\chi^2/\mathrm{ndf}$ during burn-in and production sphase.