Bayesian Calibration of the Crossterms Eigenvolume HRG Model: Integrating Lattice QCD and Experimental Data

TL;DR

This work introduces a unified Bayesian calibration of the Cross EV--HRG model, which includes flavor-dependent cross-term eigenvolume interactions, against both continuum-limit LQCD thermodynamic observables and centrality-resolved ALICE hadron yields. A Gaussian Process emulator with PCA-based dimensionality reduction enables efficient exploration of a high-dimensional parameter space, yielding full posterior distributions that reveal strong data-dependent constraints and intricate parameter correlations. The analysis demonstrates that LQCD data alone weakly constrain eigenvolumes, while hadron yields substantially sharpen constraints, particularly for baryon-related radii, with correlated systematic uncertainties playing a crucial role in reliable inference. A simultaneous LQCD+yield calibration confirms a consistent, flavor-dependent EV hierarchy and highlights that yields are the dominant source of information for the strange-baryon sector, while suggesting further work to refine the hadron spectrum and explore fluctuation observables.

Abstract

We perform a Bayesian calibration of the Cross--term Excluded-Volume Hadron Resonance Gas (Cross EV--HRG) model, which incorporates flavor-dependent repulsive interactions within a thermodynamically consistent framework. For the first time, the thermal model is simultaneously constrained using lattice QCD (LQCD) thermodynamic observables and centrality-resolved hadron yield data from Pb--Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76~\mathrm{TeV}$ measured by the ALICE Collaboration. We also find that the calibration outcome is strongly data-dependent in terms of constraining power and uncertainty structure. In particular, LQCD observables alone provide only weak constraints on the eigenvolume parameters, while the inclusion of hadron yield data substantially enhances the constraining power and induces a nontrivial reshaping of the posterior distributions. We further investigate the impact of correlated experimental systematic uncertainties by constructing a phenomenological covariance matrix and systematically varying its strength, demonstrating that a careful and consistent treatment of systematic correlations is essential for reliable parameter estimation. Across all calibration scenarios, the parameters associated with multi-strange hadrons remain only moderately constrained, which may reflect limitations of the currently established hadron resonance spectrum. No clear monotonic hierarchy of strange-hadron eigenvolume radii emerges within the present uncertainties, indicating that further dedicated studies are required.

Figures (8)

• Figure 1: Validation of the GP emulator using 50 randomly selected test parameter sets. The legend identifies the thermodynamic and particle-density observables. The lower panel shows the ratio of full-model predictions to the emulator output, demonstrating the satisfactory accuracy of the GP approximation across all observables.
• Figure 2: Total Sobol index ($S_T$) heatmaps illustrating the global sensitivity of model parameters to different observables. Left panel: Sensitivity of LQCD thermodynamic observables to species-dependent eigenvolume parameters. Each row corresponds to an observable, and each column to a specific eigenvolume parameter; horizontal blocks represent different temperature points in the LQCD dataset, ordered from low to high temperature. Right panel: Sensitivity of hadron yield observables to the model parameters, including the freeze-out temperature $T_{\mathrm{ch}}$. The color scale represents the total Sobol index on a logarithmic scale.
• Figure 3: Corner plot showing the posterior distributions and pairwise correlations of the model parameters obtained from the Bayesian calibration of the Cross EV--HRG model to LQCD data. The diagonal panels display the one-dimensional marginalized posteriors, with the posterior median, mean, and 68% credible intervals indicated in the legend. The off-diagonal panels show the corresponding two-dimensional joint posterior distributions with credible contours at the indicated levels. The color bar represents the normalized posterior density. The inset shows the evolution of $\chi^2/\mathrm{ndf}$ during the MCMC sampling, including the burn-in phase, along with the marginalized $\chi^2/\mathrm{ndf}$ distribution.
• Figure 4: Comparison between LQCD results for selected thermodynamic observables and model predictions from the Cross EV--HRG calibration. The broad green band represents model samples drawn from the pre-converged burn-in phase, while the narrower red band corresponds to predictions obtained from the marginalized posterior distribution.
• Figure 5: Same as the caption of Fig. \ref{['fig:corner_plot_lqcd']}, except that the Bayesian fit is performed using centrality-dependent hadron yield data, with experimental uncertainties treated via a diagonal covariance matrix.