A Gaussian Process Generative Model for QCD Equation of State

TL;DR

This work addresses constraining the QCD equation of state at zero net baryon density by introducing a non-parametric, physics-informed generator based on Gaussian Process Regression (GPR). It trains the GP on Hadron Resonance Gas data at low temperature and lattice QCD results at high temperature to produce random smooth cross-over EOSs $P(T)$, enforcing thermodynamic and causality constraints such as $\partial P/\partial T>0$, $\partial^2 P/\partial T^2>0$, and $0\le c_s^2\le 1/2$ (with a DNMR bound $0\le c_s^2+4/(3C_\eta)+1/C_\zeta\le 1$ using $C_\eta=C_\zeta=5$). The generated EOS are then embedded in the iEBE-MUSIC heavy-ion framework to study their impact on final-state hadronic observables and thermal photons; results show observables are highly sensitive to the speed of sound near the crossover window $T\in[0.15,0.25]$ GeV, with EOS variations propagating into multiplicities, $\langle p_T\rangle$, flow coefficients, $p_T$ fluctuations, HBT radii, and photon yields. This demonstrates the potential of a data-driven, physics-constrained EOS generator for Bayesian inference to constrain the QCD EOS and motivates future extensions to finite density. The approach yields a flexible, non-parametric alternative to fixed EOS parameterizations and provides an open-source tool for systematic EOS studies in heavy-ion phenomenology.

Abstract

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.

Figures (6)

• Figure 1: The scaled pressure $P/T^4$ for randomly generated EOSs from our GPR model (a). The other panels show $P/T^4$ (b), $s/T^3$ (c), and the square of the speed of sound $c_s^2$ (d) vs. temperature $T$ for two sets of EOS. The HotQCD + HRG EOS HotQCD:2014kolMoreland:2015dvc (the black solid line) is shown as a reference. The gray regions are used to fit the GPR.
• Figure 2: The centrality dependence of charged hadron and identified particle multiplicities (a), identified particle averaged transverse momenta $\langle p_{\rm T}\rangle$ (b), charged hadron anisotropic flow coefficients $v_n\{2\}\, (n=2, 3)$ (c), and the scaled standard deviation of charged hadron transverse momentum fluctuation (d) in Pb+Pb collisions at $\sqrt{s_\mathrm{NN}} = 5.02$ TeV using three different equations of state.
• Figure 3: The $p_{\rm T}$-differential $v_0(p_{\rm T})$ for charged hadrons (a) and protons (b) at $20-30\%$ Pb+Pb collisions at $\sqrt{s_\mathrm{NN}} = 5.02$ TeV from three different equations of state.
• Figure 4: HBT radii $R_{\rm out}$, $R_{\rm side}$, $R_{\rm long}$ and $R_{\rm out}^2-R_{\rm side}^2$ of identical pion pairs as a function of $m_{\rm T}$ for the different equations of state in mid-rapidity $0-5\%$ Pb+Pb collisions at $\sqrt{s_\mathrm{NN}} = 5.02$ TeV.
• Figure 5: Thermal photon (QGP + hot hadron gas) $p_{\rm T}$-spectra from three equations of state in $0-20\%$ Pb+Pb collisions at $\sqrt{s_\mathrm{NN}} = 5.02$ TeV.