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Equation of state, QCD phase diagram: predictions from lattice QCD

Bastian B. Brandt

TL;DR

This work surveys lattice QCD results for the QCD phase diagram and the equation of state (EoS) relevant to heavy-ion collisions. It highlights direct lattice findings at zero or limited chemical potential, including a crossover transition at $T_{pc}\sim158$ MeV and a precisely characterized EoS up to $T\sim 2$ GeV (now extended to $T\sim165$ GeV via step-scaling). It then discusses approaches to the multidimensional parameter space using indirect methods such as Taylor expansion and analytic continuation, detailing high-order coefficients and the $T'$-expansion that extend applicability to $\mu_B/T\sim3.5$ and beyond, as well as CEP-search strategies (entropy spinodals and Lee-Yang edge scaling) and their current constraints. Overall, the paper underlines the progress and remaining challenges in controlling systematic effects while expanding the accessible region in $T$–$\mu_B$ and in the presence of isospin asymmetry and external magnetic fields, with significant implications for hydrodynamic modeling and the experimental search for the QCD critical endpoint.

Abstract

I review recent results on phase structure and equation of state of strong interaction matter from lattice QCD. Particular emphasis is given to the axes where direct simulations are possible and results are obtained with sufficient control over systematic effects. I also discuss the status of approaching the region of non-zero baryochemical potentials using indirect methods.

Equation of state, QCD phase diagram: predictions from lattice QCD

TL;DR

This work surveys lattice QCD results for the QCD phase diagram and the equation of state (EoS) relevant to heavy-ion collisions. It highlights direct lattice findings at zero or limited chemical potential, including a crossover transition at MeV and a precisely characterized EoS up to GeV (now extended to GeV via step-scaling). It then discusses approaches to the multidimensional parameter space using indirect methods such as Taylor expansion and analytic continuation, detailing high-order coefficients and the -expansion that extend applicability to and beyond, as well as CEP-search strategies (entropy spinodals and Lee-Yang edge scaling) and their current constraints. Overall, the paper underlines the progress and remaining challenges in controlling systematic effects while expanding the accessible region in – and in the presence of isospin asymmetry and external magnetic fields, with significant implications for hydrodynamic modeling and the experimental search for the QCD critical endpoint.

Abstract

I review recent results on phase structure and equation of state of strong interaction matter from lattice QCD. Particular emphasis is given to the axes where direct simulations are possible and results are obtained with sufficient control over systematic effects. I also discuss the status of approaching the region of non-zero baryochemical potentials using indirect methods.

Paper Structure

This paper contains 8 sections, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Predictions from lattice QCD
  3. Relevant parameter space
  4. Direct results on the phase structure and the equation of state
  5. Towards the multi-dimensional parameter space
  6. Results for the equation of state
  7. Indirect determination of the phase structure
  8. Conclusions

Figures (3)

  • Figure 1: Phase diagram extensions in directions of $\mu_I$ (left) and magnetic fields (right). The $\mu_I$ phase diagram features a phase with a BEC of charged pions and a possible BCS phase. The phase diagram at $\mathbf{B}\neq0$ includes a critical endpoint DElia:2021yvk.
  • Figure 2: Left: Extension of the chiral crossover to $\mu_B\neq$ at $\langle n_S \rangle=0$ from Ref. HotQCD:2018pds. Right: Exclusion region for the CEP location from Ref. Borsanyi:2025dyp.
  • Figure :