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Phase structure of 2+1-flavor QCD from an Einstein-dilaton-flavor holographic model

Jin-Yang Shen, Xin-Yi Liu, Jin-Rui Wu, Yue-Liang Wu, Zhen Fang

Abstract

We construct a holographic QCD model based on the Einstein--dilaton--flavor framework with 2+1 flavors and investigate its phase structure using machine-learning techniques. At zero chemical potential, the model reproduces the equation of state and chiral transition in quantitative agreement with lattice QCD results. By varying the light and strange quark masses, we map out the quark-mass dependence of the transition order and obtain the corresponding phase diagram, which is consistent with phase structures extracted from lattice simulations and other nonperturbative approaches. In particular, the predicted first-order region is found to be small, in line with the most recent lattice QCD analyses. We also examine the critical behavior along the second-order boundaries and the tricritical region, finding that the critical exponents exhibit mean-field scaling characteristic of classical holographic constructions. Integrating machine learning with holographic QCD significantly enhances the efficiency of parameter optimization, providing a robust and practical strategy for improving the predictive power of holographic modeling of QCD thermodynamics.

Phase structure of 2+1-flavor QCD from an Einstein-dilaton-flavor holographic model

Abstract

We construct a holographic QCD model based on the Einstein--dilaton--flavor framework with 2+1 flavors and investigate its phase structure using machine-learning techniques. At zero chemical potential, the model reproduces the equation of state and chiral transition in quantitative agreement with lattice QCD results. By varying the light and strange quark masses, we map out the quark-mass dependence of the transition order and obtain the corresponding phase diagram, which is consistent with phase structures extracted from lattice simulations and other nonperturbative approaches. In particular, the predicted first-order region is found to be small, in line with the most recent lattice QCD analyses. We also examine the critical behavior along the second-order boundaries and the tricritical region, finding that the critical exponents exhibit mean-field scaling characteristic of classical holographic constructions. Integrating machine learning with holographic QCD significantly enhances the efficiency of parameter optimization, providing a robust and practical strategy for improving the predictive power of holographic modeling of QCD thermodynamics.

Paper Structure

This paper contains 20 sections, 46 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Einstein-dilaton-flavor system
  3. The bulk action
  4. The EOM
  5. The boundary condition
  6. Chiral transition and thermodynamic properties
  7. A slice of machine learning and neural network
  8. Machine learning-optimized parameters and fitting results
  9. Pseudo-critical temperatures of the QCD thermal transition
  10. Mass dependence of the QCD phase transition
  11. The tricritical point
  12. The flavor-symmetric case
  13. The critical exponents
  14. Brown segment ($m_{u,d}=0,\, m_s>m_s^{\text{tri}}$).
  15. Blue segment ($m_{u,d}\neq 0,\, m_s<m_s^{\text{tri}}$).
  16. ...and 5 more sections

Figures (17)

  • Figure 1: The quark-mass phase diagram (Columbia plot) illustrating how the QCD transition order at $\mu =0$ depends on $m_{u,d}$ and $m_s$Ding:2015ona.
  • Figure 2: Schematic diagram of the neural network architecture. The network consists of a single-input layer, three hidden layers with 128, 256, and 128 neurons, respectively, and a single-output layer.
  • Figure 3: Temperature dependence of the energy density $\varepsilon$, entropy density $s$, and pressure $p$ in the 2+1-flavor case. The results from the machine learning–optimized holographic model are shown as solid lines, while the points with error bars and shaded bands correspond to the HotQCD HotQCD:2014kol and W-B Borsanyi:2013bia lattice simulations, respectively.
  • Figure 4: Comparison of the chiral transition behavior for light (top) and strange (bottom) quarks with the HotQCD results HotQCD:2014kolGubler:2018ctz.
  • Figure 5: The subtracted chiral condensate (top) and the renormalized chiral condensate (bottom) compared with the W-B lattice results Borsanyi:2010bp.
  • ...and 12 more figures