Holographic QCD equation of state constrained by lattice QCD: neural-ODE for probe-limit and a back-reaction test

Abstract

We study the equation of state (EoS) of QCD matter in a bottom-up holographic setup that combines an Einstein-Maxwell-dilaton (EMD) sector with an improved Karch-Katz-Son-Stephanov (KKSS) flavor action. In the probe approximation, we perform an inverse reconstruction of the model functions by parameterizing them with neural networks and solving the EMD equations via a differentiable ODE solver (a neural ODE framework), calibrating the model to a $(2+1)$-flavor lattice-QCD EoS at finite temperature and finite baryon chemical potential. The reconstructed model functions are then parametrized and kept fixed across thermodynamic states. Next, viewing the EMD sector as an effective description of pure Yang--Mills theory, we fix its parameters by fitting the $μ_B=0$ lattice pure-glue EoS using a hybrid optimization strategy. Finally, we go beyond the probe limit and solve the coupled EMD$+$KKSS equations with back-reaction, using the pure-glue-calibrated EMD sector as a fixed input and varying the KKSS couplings to compare with the $μ_B=0$ two-flavor lattice EoS. We find a visible mismatch and a high-temperature behavior in which the back-reacted dimensionless ratios approach a nearly $β_1$-insensitive plateau close to the pure-glue baseline, providing a simple structural diagnostic for the present flavor-sector truncation.

Figures (9)

• Figure 1: Schematic illustration of the neural-ODE calibration workflow. For a given set of model-function parameters, the EMD equations are solved to obtain thermodynamic observables, which are compared with the reference table to construct the loss, and the parameters are updated by gradient-based optimization.
• Figure 2: Probe approximation, $(2+1)$ flavors at $\mu=0$: comparison between the holographic EMD results (solid lines) and the reference thermodynamic table Abuali:2025tbd (scatter points). Upper left panel: $s/T^3$; upper right panel: $p/T^4$; lower left panel: $\epsilon/T^4$; lower right panel: $I/T^4=(\epsilon-3p)/T^4$. The reference points for $I/T^4$ are constructed from the combination $\epsilon-3p$ and thus do not represent an independently tabulated observable. The horizontal axis is $T$ in units of MeV.
• Figure 3: Probe approximation, $(2+1)$ flavors in fixed $\mu/T$ slices (temperature window $T=35$--$490~\mathrm{MeV}$): comparisons between the holographic EMD results (solid lines) and the reference thermodynamic table Abuali:2025tbd (scatter points). Upper left panel: $s/T^3$; upper right panel: $p/T^4$; lower left panel: $\epsilon/T^4$; lower right panel: $n_B/T^3$.
• Figure 4: Probe calibration via neural ODE: reconstructed model functions $V_{\phi}(\phi)$ (upper panel) and $h_{\phi}(\phi)$ (lower panel). The shaded region indicates the range of $\phi$ values effectively covered by the input thermodynamic data at $\mu=0$ (in the present dataset, roughly $\phi\simeq 1.1$--$7.0$), which provides a practical measure of the domain where the reconstruction is best constrained.
• Figure 5: Determination of the QCD critical endpoint (CEP) in the probe-approximation model. Upper panel: $T(s,\mu_B)$ curve with the CEP marked by a red dot. Lower panel: $\mathrm{d}T/\mathrm{d}s$ curve, where the extremum (black star) corresponds to the CEP.