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

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.

💡 Research Summary

The review by Bastian B. Brandt provides a comprehensive overview of the current status of lattice QCD calculations of the QCD phase diagram and the equation of state (EoS) of strongly interacting matter, with a focus on regions where direct Monte‑Carlo simulations are feasible and on the systematic control of uncertainties. The paper first outlines the importance of non‑perturbative methods for interpreting heavy‑ion collision experiments, emphasizing that lattice QCD remains the only approach that can be systematically improved. Direct simulations are possible at vanishing baryon chemical potential (µ_B=0) and along axes corresponding to non‑zero isospin chemical potential (µ_I) or external magnetic fields (B). In the µ_I direction, state‑of‑the‑art 2+1 flavor simulations have mapped out a phase diagram featuring a Bose‑Einstein condensate of charged pions for µ_I≥m_π/2, a pseudo‑tricritical point where the chiral crossover meets the second‑order BEC line, and ongoing searches for a BCS‑type superconducting phase at larger µ_I. For homogeneous magnetic fields, continuum extrapolated results show that the crossover temperature decreases monotonically with |eB|, the transition sharpens, and a critical endpoint appears for |eB| between 4 and 9 GeV². Both axes have yielded high‑precision EoS data up to temperatures of order 2 GeV, with recent step‑scaling methods extending the reach to ~165 GeV.

The bulk of the review discusses indirect methods required to explore the µ_B≠0 region, where the sign problem prevents direct simulations. The Taylor‑expansion approach expands the pressure in powers of µ_X/T, with coefficients χ_BQS^{ijk} computed at µ=0. Current lattice ensembles (Nt=8) have produced coefficients up to tenth order in small volumes (LT=2) and eighth order in larger volumes (LT=4); only the fourth‑order coefficients are available in the continuum limit, while sixth‑order results show tension that may disappear after continuum extrapolation. Using the latest eighth‑order coefficients, the EoS is reliable up to µ_B/T≈2–3, and the newer T′‑expansion technique pushes the applicability to µ_B/T≈3 or beyond. Analytic continuation from imaginary µ_B, combined with Padé approximants, provides complementary estimates. Extensions to include magnetic fields and isospin chemical potentials have been performed via second‑order Taylor expansions and analytic continuation, revealing non‑monotonic behavior of thermodynamic observables at large B and identifying χ_BQS^{110} as a sensitive “magnetometer”.

The review also surveys recent strategies to locate the conjectured critical endpoint (CEP). Entropy spinodal analysis (critical lensing) originally suggested a CEP at (T≈114 MeV, µ_B≈600 MeV), but more refined studies have excluded a CEP up to µ_B≈450 MeV. Lee‑Yang zero scaling, which tracks the complex µ_B singularity closest to the origin, has produced tentative CEP signals around (T≈105 MeV, µ_B≈420 MeV) using eighth‑order Taylor data or Padé fits, yet high‑precision work highlights large systematic uncertainties, yielding only an upper bound T_CEP≲103 MeV. The author emphasizes that while direct results along the µ_I and B axes are robust, the µ_B plane still suffers from methodological limitations, and future progress hinges on reducing systematic errors, achieving continuum extrapolations for higher‑order Taylor coefficients, and developing more reliable analytic continuation or alternative approaches.

In conclusion, lattice QCD has delivered precise predictions for the QCD equation of state and phase structure in regions amenable to direct simulation, and indirect methods have steadily expanded the accessible µ_B range. However, definitive statements about the existence and location of a critical endpoint remain elusive. Continued improvements in computational techniques, larger and finer lattices, and innovative analysis methods are essential for delivering the quantitative input required by heavy‑ion phenomenology and neutron‑star physics.