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 $\mu_B=0$ 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 $\mu_B=0$ two-flavor lattice EoS. We find a visible mismatch and a high-temperature behavior in which the back-reacted dimensionless ratios approach a nearly $\beta_1$-insensitive plateau close to the pure-glue baseline, providing a simple structural diagnostic for the present flavor-sector truncation.