Finite-Temperature QCD Equation of State

Continuum-extrapolated lattice calculations support a quantitatively consistent zero-density QCD equation of state through the crossover and into the several-hundred-MeV plasma regime. HotQCD and Wuppertal–Budapest reach compatible thermodynamics using different staggered actions and analysis choices. This agreement does not remove the sign problem at finite baryon density, and the newest multi-GeV calculation uses three massless flavors rather than the physical crossover theory.

Evidence cutoff. 11 August 2026. Reassess by 11 February 2027 or after a new continuum dataset changes the shared temperature range, flavor content, or covariance.

Required background. QCD thermodynamics and the equation of state defines the observables and heavy-ion interface; lines of constant physics and continuum extrapolation supplies the regulator-removal test.

Helpful background. QCD phase structure distinguishes the zero-density crossover from a critical point; algorithm validation and ensemble provenance identifies sampling dependencies; complete lattice error budgets organizes correlated systematics; QCD fields and the perturbative domain controls the high-temperature comparison.

The thermodynamic claim

At baryon chemical potential $\mu_B=0$, the equation of state may be represented by pressure $p(T)$, energy density $e(T)$, entropy density $s(T)$, and the interaction measure

$$I(T)=e(T)-3p(T).$$

The central claim is agreement among continuum-extrapolated determinations of these quantities for physical $2+1$-flavor QCD over their shared temperature range, especially $130\text{–}400\ \mathrm{MeV}$. “Agreement” means compatible continuum curves within declared uncertainties after accounting for differing temperature scales and normalizations. This brief does not assess nonzero $\mu_B$, magnetic fields, transport coefficients, or a heavy-ion inversion of the equation of state.

Evidence matrix

Source and method	Relation to the bounded claim	Independence	Result and stated uncertainty	Main limitation
HotQCD, 2014, HISQ/tree ensembles with $N_\tau=6,8,10,12$	supports a continuum physical-mass equation of state	distinct action, ensembles, scale analysis, and collaboration from Wuppertal–Budapest	continuum thermodynamics for roughly $130\text{–}400\ \mathrm{MeV}$; compatible with Wuppertal–Budapest in the shared region	correlations among temperature points and scale setting complicate pointwise comparisons
Wuppertal–Budapest, 2014, stout staggered ensembles through $N_\tau=16$	independently supports the crossover equation of state	different discretization and analysis; common QCD parameters and some external scale inputs remain	physical $2+1$-flavor continuum curves and public tables, consistent with HotQCD	compact publication gives less reusable covariance than a full likelihood record
Bresciani et al., 2025, shifted-boundary lattice calculation	extends nonperturbative control toward electroweak temperatures	different Wilson discretization and entropy method	$N_f=3$ massless-QCD equation of state from $3$ to $165\ \mathrm{GeV}$ at about $0.5\%$–$1.0\%$ precision	not the same flavor/mass theory as physical $2+1$ QCD near the crossover; not a direct replication of the 2014 curves
Hadron-resonance-gas and resummed perturbative comparisons in HotQCD, 2014	qualify the low- and high-temperature limits	analytic comparisons reuse QCD inputs but not Monte Carlo ensembles	smooth matching trends outside the peak-interaction region	model spectrum at low $T$ and perturbative truncation at high $T$ prevent either from replacing the lattice result

Agreement and remaining systematics

The two 2014 programs are genuinely complementary: their lattice actions, improvement schemes, temporal extents, and some scale choices differ. Their continuum curves’ agreement in the shared physical-mass range is therefore stronger than agreement between two fits to the same ensembles. It supports the practical equation of state used as an input to many zero-density hydrodynamic models.

The effective number of checks is still smaller than the paper count. Temperature calibration, physical hadron inputs, taste-breaking corrections, the integration used to recover pressure, and perturbative normalization can introduce long-range correlations across temperatures. Tables without full covariance should not be converted into dozens of independent agreeing points.

The 2025 high-temperature calculation tests a different and valuable limit. Its shifted-boundary entropy method and Wilson regulator reduce dependence on the crossover analyses, but its three massless flavors are a controlled high-energy approximation, not a new physical-mass determination at $T\sim155\ \mathrm{MeV}$. Flavor thresholds must be matched before using it in a full Standard Model cosmological equation of state.

What this evidence does not establish

• Zero-density agreement does not solve the finite-density sign problem or locate a QCD critical point.
• A smooth crossover in equilibrium does not determine real-time viscosities or hydrodynamization.
• Continuum extrapolation removes the lattice-spacing regulator only within the tested fit and temperature range; it does not erase finite-volume, mass-tuning, or scale errors.
• The high-temperature $N_f=3$ calculation cannot be spliced directly onto physical $2+1+1$ thermodynamics without threshold and scheme matching.

What would change the assessment

The claim would strengthen with public correlated continuum data from action-distinct calculations, physical charm included across the overlap region, and a continuous matching chain from the crossover to the multi-GeV calculation. It would weaken if a finer-lattice program found a stable continuum curve outside the present combined uncertainties or if scale-setting changes moved the crossover thermodynamics coherently.

For phenomenology, the most useful next product is not another smooth parameterization alone but a versioned table with covariance, scale convention, flavor content, and interpolation error that can be propagated through hydrodynamic inference.

Source selection

The finite set contains the two independent physical-mass continuum programs that define the crossover benchmark and the 2025 calculation that extends nonperturbative coverage to much higher temperature. Older coarse-lattice results, noncontinuum model curves, finite-density Taylor expansions, and phenomenological inversions were excluded because they answer different questions.

Related research pages

• Euclidean lattice fields and continuum extrapolation for the shared regulator and error logic.
• Finite-density QCD in controlled regimes for the domain deliberately excluded here.
• QCD critical point: existence and location for the distinct finite-density claim.

References

• Bazavov, A., et al. (HotQCD Collaboration). “Equation of State in $(2+1)$-Flavor QCD.” Physical Review D 90 (2014): 094503. DOI; arXiv.
• Borsányi, Szabolcs, et al. “Full Result for the QCD Equation of State with $2+1$ Flavors.” Physics Letters B 730 (2014): 99–104. DOI; open record and data links.
• Bresciani, Matteo, Mattia Dalla Brida, Leonardo Giusti, and Michele Pepe. “QCD Equation of State with $N_f=3$ Flavors up to the Electroweak Scale.” Physical Review Letters 134 (2025): 201904. DOI; arXiv.