Machine learning holographic black hole from lattice QCD equation of state - Extracts CEP using EMD framework

Aug 5, 2026

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Based on lattice QCD equation of state and baryon number susceptibility at zero baryon chemical potential, this work uses machine learning together with the Einstein-Maxwell-Dilaton (EMD) framework to build an analytic holographic black hole metric for pure gluon, 2-flavor, and 2+1-flavor systems. Dilaton potentials reconstructed from Einstein equations agree with extended DeWolfe-Gubser-Rosen type potentials fixed by lattice data, supporting the framework’s robustness. The 2+1-flavor critical endpoint is predicted at Tc = 0.094 GeV and μBc = 0.74 GeV.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages6

Reading Time15 min

AccessFree to download

Introduction

The general EMD framework

Action and fields

Metric ansatz and boundary conditions

Machine learning-assisted metric reconstruction

Predictions for the critical endpoint (CEP)

arXiv :2401 .06417v2 [hep-ph] 30 Mar 2024 Machine learning holographic black hole from lattice QCD equation of state Xun Chen∗ School of Nuclear Science and Technology, University of South China, Hengyang 421001, China Mei Huang† School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China (Dated: April 2, 2024) Based on lattice QCD results of equation of state (EOS) and baryon number susceptibility at zero baryon chemical potential, and supplemented by machine learning techniques, we construct the analytic form of the holographic black hole metric in the Einstein-Maxwell-Dilaton (EMD) framework for pure gluon, 2-􀀍avor, and 2+1-􀀍avor systems, respectively. The dilaton potentials solved from Einstein equations are in good agreement with the extended non-conformal DeWolfe-GubserRosen (DGR) type dilaton potentials 􀀌xed by lattice QCD EOS, which indicates the robustness of the EMD framework. The predicted critical endpoint (CEP) in the 2+1-􀀍avor system is located at (Tc =0.094GeV, 􀀖cB=0.74GeV), which is close to the results from the realistic Polyakov-NambuJona-Lasinio(PNJL) model, the functional renormalization group, and the holographic model with extended DeWolfe-Gubser-Rosen dilaton potential. Introduction: Exploring phase transitions and phase structures of Quantum Chromodynamics (QCD) matter under extreme conditions is essential for understanding phenomena in heavy ion collisions, the early universe, and neutron stars. It has been predicted that a critical endpoint (CEP) exists at a 􀀌nite baryon chemical potential µB [1], and it has attracted extensive attention for several decades both in theory and experiment[2–7] . Searching for the CEP has become one of the most important goals at high baryon densities in heavy ion collisionsat relativistic heavy ion collision (RHIC) [8–13], as well as in future facilities, e.g., FAIR at Darmstadt, NICA in Dubna and HIAF in Huizhou. Due to the sign problem, lattice QCD is not well adapted to 􀀌nite chemical potential regions. The CEPhas been extensively investigated in 4-dimension e􀀋ective QCD models, e.g., the Nambu-Jona-Lasinio (NJL), linear sigma model[14 , 15], and their Polyakov-loop extended version [16–19], the Dyson-Schwinger equations (DSE) [20–23], and the functional renormalization group (FRG)[24, 25] . In recent decades, the holographic gauge-gravity duality [26] has been widely applied as an important nonperturbative method in describing hadron physics [27–30] and QCD matter under extreme conditions [31–48] . The 5-dimensional EinsteinMaxwell-Dilaton (EMD) framework [34–39, 45–48] has been adapted as the working framework for describing QCD matter at 􀀌nite temperature and density. A family of 􀀌ve-dimensional black holes dual to QCD equation of state has been constructed in Refs. [31, 49] with a non-conformal dilaton potential, and a CEP was 􀀌rstly obtained from holographic dual black hole by DeWolfe-Gubser-Rosen (DGR) in [32] . Further careful studies have been conducted with extended DGR non-conformal dilaton potential with more parameters 􀀃 † [chenxun@usc.edu.cn](chenxun@usc.edu.cn) [huangmei@ucas.ac.cn](huangmei@ucas.ac.cn); corresponding author [42, 43 , 50–53], see review in [54] . Another equivalent method is the potential reconstruction method, where one can input the dilaton or a metric to determine the dilaton potential. Although this approach results in a temperature-dependent dilaton potential, the model can still capture many QCD properties through analytical solutions. Machine learning has become a useful tool in highenergy physics; for a recent review, see Ref. [55] . Furthermore, the integration of deep learning with holographic QCD has been explored in recent studies[56–63] . Unlike conventional holographic models, this approach 􀀌rst employs speci􀀌c QCD data to determine the bulk metric (as well as other model parameters) through machine learning. Subsequently, the model utilizes the

FAQs

What inputs from lattice QCD are used in constructing the holographic model?

The model is built from lattice QCD results for the equation of state and baryon number susceptibility at zero baryon chemical potential.

How does the work obtain the holographic black hole metric and dilaton potential?

It combines a potential reconstruction method with machine learning to extract the EMD bulk metric from lattice EOS data, then determines dilaton potentials via Einstein equations.

Where is the predicted critical endpoint located in the 2+1-flavor system?

The predicted critical endpoint is at Tc = 0.094 GeV and μBc = 0.74 GeV.

Aug 5, 2026

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