Data-Driven Refinement of an Analytical Holographic Model for the QCD Phase Transition

Abstract

Using (2+1)-flavor lattice QCD data, we refine the parameters of an analytical holographic model via gradient descent optimization to precisely locate the critical endpoint in the $T-μ$ plane. Specifically, we calibrate the model using input data for the speed of sound at $μ_B = 0$, the second-order baryon number susceptibility $χ^B_2$, and the baryon number density at $μ_B/T = 1$. With these parameters fixed, we calculate pressure and energy density versus temperature at small chemical potentials and compare the results with lattice QCD data using Taylor expansion techniques. This comparison validates the robustness of our model upon extension to finite chemical potentials, as the results show broad consistency with lattice QCD data in this regime. Finally, we employ the calibrated model to determine the coordinates of the critical endpoint in the $T-μ_B$ plane, finding it located at $(μ_B = 0.678 \, \rm{GeV}, T = 0.110 \, \rm{GeV})$

Figures (6)

• Figure 1: Comparison of (2+1)-flavor lattice QCD HotQCD:2014kolHotQCD:2012fhjBazavov:2017dus and holographic model predictions for: (a) thermodynamic quantities (entropy $s$, energy $\epsilon$, pressure $p$, trace anomaly $\epsilon - 3p$) versus temperature at vanishing chemical potential; (b) second-order baryon number susceptibility versus temperature at vanishing chemical potential.
• Figure 2: Comparison of (2+1)-flavor lattice QCD Bollweg:2022rps and holographic model predictions for: (a) baryon number density versus temperature at finite chemical potential; (b) second-order baryon number susceptibility versus temperature at finite chemical potential.
• Figure 3: Comparison between (2+1)-flavor lattice QCD HotQCD:2014kol and holographic model predictions at zero chemical potential for: (a) temperature dependence of the squared speed of sound; (b) temperature dependence of the specific heat.
• Figure 4: (a) Comparison of pressure between our holographic model and lattice QCD data at various $\mu/T$ values Bazavov:2017dus; (b) Comparison of energy density $\epsilon$ between model predictions and lattice results at various $\mu/T$ values Bazavov:2017dus.
• Figure 5: (a) The QCD phase diagram in the $T$-$\mu$ plane; (b) Comparison of our model predictions with those of other theoretical models, including: the CEP-excluded region from lattice QCD Borsanyi:2025dyp; the CEP position from Lattice QCD ($\mu_{B}^{\mathrm{CEP}} = 422_{-35}^{+80}~\mathrm{MeV}$, $T^{\mathrm{CEP}} = 105_{-18}^{+8}~\mathrm{MeV}$) Clarke:2024ugt; from V-QCD ($\mu_{B}^{\mathrm{CEP}} = 626_{-179}^{+90}~\mathrm{MeV}$, $T^{\mathrm{CEP}} = 105_{-6}^{+14}~\mathrm{MeV}$) Ecker:2025vnb; from LYE-DSE Wan:2024xeu; and from FRG Fu:2023lcm.