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Thermal field theory and the QCD Equation of State

Matteo Bresciani

TL;DR

This chapter develops a comprehensive framework for QCD thermodynamics at finite temperature and density. It combines the Euclidean path-integral formulation with a thermal effective theory program that separates hard, soft, and ultrasoft scales into EQCD and MQCD, enabling a controlled treatment of the QCD pressure and the Equation of State. The discussion covers perturbative approaches (including HTL resummation) and non-perturbative lattice QCD determinations, detailing how the pressure and trace anomaly inform the thermodynamics and cosmological implications. It also surveys the QCD phase diagram at zero and nonzero chemical potential, highlighting the crossover nature at μ=0, the Columbia plot landscape, and the ongoing search for a possible critical point at finite density. The work provides a unified, multi-method toolkit for predicting the EoS across wide ranges of temperature and density, with direct relevance to heavy-ion phenomenology and early Universe cosmology, while noting key open questions in high-temperature/density QCD.

Abstract

This Chapter introduces QCD at finite temperature and density. We first present the formulation of the thermal theory in the Euclidean path integral formalism. We then describe how the strong dynamics at high temperature can be inspected through thermal effective field theories. As a concrete example of thermodynamic quantity, we discuss the Equation of State, which characterises the equilibrium properties of the QCD plasma. We finally conclude with an overview of the phase diagram of strongly interacting matter.

Thermal field theory and the QCD Equation of State

TL;DR

This chapter develops a comprehensive framework for QCD thermodynamics at finite temperature and density. It combines the Euclidean path-integral formulation with a thermal effective theory program that separates hard, soft, and ultrasoft scales into EQCD and MQCD, enabling a controlled treatment of the QCD pressure and the Equation of State. The discussion covers perturbative approaches (including HTL resummation) and non-perturbative lattice QCD determinations, detailing how the pressure and trace anomaly inform the thermodynamics and cosmological implications. It also surveys the QCD phase diagram at zero and nonzero chemical potential, highlighting the crossover nature at μ=0, the Columbia plot landscape, and the ongoing search for a possible critical point at finite density. The work provides a unified, multi-method toolkit for predicting the EoS across wide ranges of temperature and density, with direct relevance to heavy-ion phenomenology and early Universe cosmology, while noting key open questions in high-temperature/density QCD.

Abstract

This Chapter introduces QCD at finite temperature and density. We first present the formulation of the thermal theory in the Euclidean path integral formalism. We then describe how the strong dynamics at high temperature can be inspected through thermal effective field theories. As a concrete example of thermodynamic quantity, we discuss the Equation of State, which characterises the equilibrium properties of the QCD plasma. We finally conclude with an overview of the phase diagram of strongly interacting matter.
Paper Structure (16 sections, 27 equations, 3 figures)

This paper contains 16 sections, 27 equations, 3 figures.

Table of Contents

  1. Thermal field theory and the QCD Equation of State
  2. Introduction
  3. QCD thermodynamics
  4. Thermal effective theory of QCD
  5. Electrostatic QCD
  6. Magnetostatic QCD
  7. The QCD pressure
  8. Scope of the effective theory
  9. QCD Equation of State
  10. Perturbative methods
  11. Non-perturbative determination
  12. Finite density
  13. Thermal phases of QCD
  14. The $\mu=0$ case
  15. The $\mu\neq0$ case
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Equation of State of QCD with three flavours. Left panel: results for the trace anomaly, pressure and entropy density in the interval $130-400$ MeV determined in Refs. Borsanyi:2013biaHotQCD:2014kol. Plot taken from Ref. HotQCD:2014kol. Right panel: results for the entropy density, energy density and pressure for temperatures up to the electroweak scale Bresciani:2025vxwBresciani:2025mcu. The value at 500 MeV is taken from Refs. Borsanyi:2013biaBazavov:2017dsy. The dashed lines represent the Stefan-Bolzmann value of the three thermodynamic functions.
  • Figure 2: Left panel: the non-perturbative results for the entropy density (black dots) from Refs. Bresciani:2025vxwBresciani:2025mcu are compared to standard perturbation theory. Each curve includes up to the order in the strong coupling (here denoted by $\hat{g}$) reported by the label. Shadowed bands correspond to a variation of the renormalization scale by the factors $2$ or $1/2$. Data are represented as a function of $\hat{g}$, and some values of the physical temperature are reported for reference. Right panel: the entropy density is compared to HTL perturbation theory at leading order (LO), next-to leading order (NLO) and next-to-next-to leading order (NNLO). Shadowed bands have the same meaning as in the left panel.
  • Figure 3: Left: Columbia plot from Ref. Philipsen:2010gj. Right: conjectured phase diagram of QCD.