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QCD phase structure & equation of state: A functional perspective

Fabian Rennecke

TL;DR

The paper surveys how first-principles functional methods (DSEs and FRG) can map the QCD phase structure at finite density, where lattice methods struggle. It reports a CEP location near temperatures around 110 MeV and baryon chemical potential around 630 MeV, the possible existence of a moat regime and inhomogeneous phases, and a robust, self-consistent equation of state extending to large densities with a notable color-superconducting gap. The results provide qualitative guidance for heavy-ion experiments and neutron-star physics, and highlight the need for systematic error control and exploration of spatially modulated phases. Overall, functional QCD offers a principled route to connect microscopic quark-gluon dynamics with macroscopic thermodynamics in regimes inaccessible to conventional lattice approaches.

Abstract

The phase structure of QCD remains an open fundamental problem of standard model physics. In particular at finite density, our knowledge is limited. Yet, numerous model studies point towards a rich and complex phase diagram at large density. Functional methods like the functional renormalization group and Dyson-Schwinger equations offer a way to study hot and dense QCD matter directly from first principles. I will discuss the phase structure of QCD and its experimental signatures through the lens of these methods.

QCD phase structure & equation of state: A functional perspective

TL;DR

The paper surveys how first-principles functional methods (DSEs and FRG) can map the QCD phase structure at finite density, where lattice methods struggle. It reports a CEP location near temperatures around 110 MeV and baryon chemical potential around 630 MeV, the possible existence of a moat regime and inhomogeneous phases, and a robust, self-consistent equation of state extending to large densities with a notable color-superconducting gap. The results provide qualitative guidance for heavy-ion experiments and neutron-star physics, and highlight the need for systematic error control and exploration of spatially modulated phases. Overall, functional QCD offers a principled route to connect microscopic quark-gluon dynamics with macroscopic thermodynamics in regimes inaccessible to conventional lattice approaches.

Abstract

The phase structure of QCD remains an open fundamental problem of standard model physics. In particular at finite density, our knowledge is limited. Yet, numerous model studies point towards a rich and complex phase diagram at large density. Functional methods like the functional renormalization group and Dyson-Schwinger equations offer a way to study hot and dense QCD matter directly from first principles. I will discuss the phase structure of QCD and its experimental signatures through the lens of these methods.

Paper Structure

This paper contains 5 sections, 1 equation, 5 figures.

Table of Contents

  1. Introduction
  2. Functional methods
  3. Phase transitions
  4. The moat regime
  5. EoS of dense QCD matter

Figures (5)

  • Figure 1: Left: Kurtosis of the net-baryon distribution from functional methods Fu:2023lcmLu:2025cls together with the kurtosis of net-protons measured by STAR Lu:2025cls. Right: Determination of the size of the chiral $O(4)$ scaling region though the critical exponent $\delta$Braun:2023qak.
  • Figure 2: Left: Static pion energy just outside (solid red line) and inside the moat regime Fu:2024rto, cf. Fig. \ref{['fig:PD']}. Right: Phase diagram of a rainbow-ladder QCD model in the chiral limit, featuring an inhomogeneous instability at small $T$ and large $\mu$ (yellow region) Motta:2024rvk.
  • Figure 3: Left: Entropy density from functional QCD in comparison to lattice results at low and intermediate baryon density Lu:2025cls. Right: Speed of sound squared at large density in comparison to chiral effective field theory and weak-coupling results Leonhardt:2019fuaBraun:2021uuaGeissel:2024nmx.
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