A Gaussian Process Generative Model for QCD Equation of State

We develop a generative model for the nuclear matter equation of state at zero net baryon density using the Gaussian Process Regression method. We impose first-principles theoretical constraints from lattice QCD and hadron resonance gas at high- and low-temperature regions, respectively. By allowing the trained Gaussian Process Regression model to vary freely near the phase transition region, we generate random smooth cross-over equations of state with different speeds of sound that do not rely on specific parameterizations. We explore a collection of experimental observable dependencies on the generated equations of state, which paves the groundwork for future Bayesian inference studies to use experimental measurements from relativistic heavy-ion collisions to constrain the nuclear matter equation of state.

💡 Research Summary

The authors present a novel, non‑parametric approach to generate QCD equations of state (EOS) at zero net baryon chemical potential using Gaussian Process Regression (GPR). By training the GPR on first‑principles data—low‑temperature results from the Hadron Resonance Gas (HRG) model and high‑temperature results from lattice QCD—they enforce reliable behavior in the well‑constrained regions while allowing the model to freely explore the crossover region (≈150–250 MeV). The GP is built for the logarithm of the scaled pressure, ln (P/T⁴), as a function of ln (T/1 GeV) with a squared‑exponential kernel (σ = 1, optimized length scale l ≈ 0.487). After training, random samples are drawn from the posterior, yielding many candidate P(T) curves. Each candidate is filtered through strict thermodynamic and causality constraints: positive first and second temperature derivatives of pressure, and a speed‑of‑sound squared 0 ≤ c_s² < 1 (the authors adopt c_s²_max = ½ for demonstration). This filtering produces a physically admissible ensemble of EOS, exemplified by 100 curves shown in Fig. 1a. Two representative EOS (labeled EOS 1 and EOS 2) are selected for detailed study; EOS 1 exhibits a lower scaled pressure and a pronounced peak in c_s² near the crossover, while EOS 2 has higher pressure and a suppressed c_s² in the same region.

The generated EOS are converted to P(e) tables and fed into the iEBE‑MUSIC framework, which couples IP‑Glasma initial conditions, second‑order DNMR viscous hydrodynamics, and a hadronic afterburner (UrQMD). Because bulk‑viscous relaxation time τ_Π traditionally depends on (1/3 − c_s²), the authors modify this parametrization to remain well‑defined when c_s² ≥ 1/3. They also adjust the coupling λ_Ππ to its maximal allowed value, ensuring linearized causality (0 ≤ c_s² ≤ 8/15) for the chosen transport coefficients (C_η = C_ζ = 5).

Large‑scale event‑by‑event simulations are performed for central and peripheral Au+Au (or Pb+Pb) collisions. Observables examined include charged‑hadron yields, identified particle spectra, mean transverse momentum ⟨p_T⟩, and anisotropic flow coefficients v_n. The results display a clear ordering that mirrors the c_s² hierarchy in the crossover temperature window: EOS 1 (largest c_s²) yields the highest ⟨p_T⟩ and the strongest v₂, reflecting more vigorous radial acceleration (D_μ ≈ c_s² ∇_μ e/e). EOS 2 (smallest c_s²) produces softer spectra and weaker flow, while the lattice‑QCD based EOS sits in between. The p/π ratio is also sensitive, with EOS 1 showing an enhanced ratio due to stronger bulk‑viscous corrections at particlization.

The study demonstrates that a GPR‑based generative EOS provides a flexible prior that respects fundamental physics while capturing a wide range of possible speed‑of‑sound profiles. This flexibility is crucial for forthcoming Bayesian inference efforts that aim to constrain the QCD EOS directly from experimental data. The authors outline future extensions to finite baryon chemical potential, incorporation of kernels that can model first‑order transitions, and systematic Bayesian calibration using the generated EOS ensemble.