The Equation of State of QCD up to very high temperatures

Leonardo Giusti; Matteo Bresciani; Mattia Dalla Brida; Michele Pepe

REVIEW 2 major objections 3 minor 38 references

The Equation of State of QCD up to very high temperatures

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper reports the first non-perturbative lattice computation of the QCD entropy density from 3 GeV to 165 GeV, with 0.5–1.0% accuracy.

desk verdict First non-perturbative Nf=3 QCD entropy density from 3 to 165 GeV; credible and useful, with a continuum-extrapolation caveat that needs the companion paper's full error budget. read the letter →

arxiv 2502.04239 v1 pith:HYJVQ5RC submitted 2025-02-06 hep-lat

Matteo Bresciani , Mattia Dalla Brida , Leonardo Giusti , Michele Pepe This is my paper · ORCID

classification hep-lat

keywords QCDequationofstateentropydensityshiftedboundaryconditionslatticeathightemperaturenon-perturbativethermodynamicsSchrödingerfunctionalcouplingcontinuumextrapolationearlyUniversecosmology

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The authors report a non-perturbative lattice QCD computation of the entropy density of the quark-gluon plasma at nine temperatures between 3 GeV and 165 GeV, with $N_f = 3$ massless quark flavours. They claim this is the first non-perturbative equation-of-state result in this range, at 0.5–1.0% precision. The calculation works in a moving reference frame with shifted boundary conditions, so the entropy density is obtained as the derivative of the free-energy density with respect to the boundary shift, which removes the power divergences and the need for a zero-temperature subtraction. The continuum results are compared with high-temperature perturbation theory and combined with lower-temperature lattice data to give a parametrization of $s/T^3$ for $T \ge 500$ MeV at about 1% relative error. If correct, this provides a first-principles input for early-Universe cosmology and heavy-ion phenomenology across almost two orders of magnitude in temperature.

What carries the argument

The load-bearing object is thermal QCD formulated with shifted boundary conditions in a moving frame: along the compact Euclidean time direction the fields close with a spatial shift, $A_\mu(x_0+L_0,\mathbf{x}) = A_\mu(x_0,\mathbf{x}-L_0\boldsymbol{\xi})$ and likewise for the fermion fields, so the system is at temperature $T = 1/(L_0\sqrt{1+\boldsymbol{\xi}^2})$ with spatial volume $V_s = L^3/\sqrt{1+\boldsymbol{\xi}^2}$. Euclidean Lorentz invariance then gives the identity $s/T^3 = \frac{1+\boldsymbol{\xi}^2}{\xi_k} \frac{1}{T^4} \frac{\partial f_{\boldsymbol{\xi}}}{\partial \xi_k}$, which carries the argument because the entropy density is a derivative of the free energy with respect to the shift: a physical quantity free of ultraviolet power divergences and, on the lattice, free of the usual zero-temperature subtraction. The numerical evaluation uses the same identity on the lattice, splitting the free-energy derivative into a fermionic piece obtained by integrating the shifted chiral condensate in the bare quark mass and a pure-gauge piece obtained by integrating the shifted plaquette action in the bare coupling; the lines of constant physics are fixed by the non-perturbative running of the Schrödinger functional coupling. The continuum limit is controlled by the fit ansatz $s(T_i,a/L_0)/T_i^3 = s(T_i)/T_i^3 + d_2 (a/L_0)^2 g_i^3 + d_3 (a/L_0)^3 g_i^3$.

What would settle it

Repeat the entropy-density computation at a fifth, finer lattice spacing (for example $L_0/a = 12$) at two or three of the reported temperatures, keeping the same lines of constant physics; if the continuum-extrapolated $s/T^3$ moves by more than the quoted $0.5$–$1.0\%$ errors, or if the data visibly deviate from the assumed $a^2$ scaling, the central claim is wrong. A cross-check independent of the fit ansatz would be to compute $s/T^3$ from the renormalized energy-momentum tensor one-point functions in the same shifted-boundary formulation and compare at a fixed temperature.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is the continuum-extrapolated, non-perturbative entropy density of $N_f=3$ QCD: $s/T^3 = 20.13(8)$ at $T = 165$ GeV and $19.58(17)$ at $T = 3.04$ GeV, with a relative error of $0.5$–$1.0\%$. These values come from a direct lattice evaluation of $s/T^3$ as the derivative of the free-energy density with respect to the shift in the boundary conditions, after one-loop improvement and extrapolation from four lattice spacings $L_0/a = 4, 6, 8, 10$. The authors further show that the temperature dependence in this range is described by a simple third-order polynomial in the five-loop $\overline{\rm MS}$ coupling $\hat g(T)$, and that the fixed-order perturbative expansion of the pressure does not reproduce the data; only a fit that absorbs unknown $O(\hat g^6)$ and higher coefficients matches them. By including known $T = 500$ MeV lattice results, they obtain a parametrization of $s/T^3$ for $T \ge 500$ MeV with a relative error below about $1\%$.

Load-bearing premise

The load-bearing assumption is that, after the one-loop improvement, the remaining discretization errors at the three finest lattice spacings are exactly the $a^2$ term in Eq. (14) plus an $a^3$ term whose size is estimated from the coarsest lattice spacing; if unmodeled higher-order or logarithmic cutoff effects shift the $L_0/a = 6, 8, 10$ data, the continuum values would move outside the quoted $0.5$–$1.0\%$ errors.

Editorial extensions

If this is right

The pressure $p(T)$ and energy density $e(T)$ follow from the tabulated entropy density through $s = dp/dT$ and $Ts = e+p$, giving a complete non-perturbative equation of state over the whole 3–165 GeV range.
The 0.5–1.0% accurate data provide a sharp benchmark for high-temperature perturbation theory; the paper shows that fixed-order perturbation theory is insufficient and that an extended fit absorbing unknown $O(\hat g^6)$ and $O(\hat g^7)$ terms is required up to the electroweak scale.
The same strategy can be generalized to four and five (massive) quark flavours, which would upgrade the Standard Model equation of state used in cosmological calculations of primordial gravitational-wave spectra and dark-matter relic abundances.
Because the moving-frame method sets the renormalization scale near each simulated temperature, it overcomes the scale-window problem that previously made multi-GeV lattice thermodynamics impractical.
The parametrization of $s/T^3$ for $T \ge 500$ MeV at about 1% relative error means the QCD contribution to the equation of state in this region can be treated as a controlled input in early-Universe calculations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension, not pursued here, is to compute other thermodynamic response functions such as the specific heat or speed of sound from the same shifted-boundary derivatives; they should carry the same 0.5–1.0% precision.
The fitted slope in $\hat g^2$ differs from the perturbative coefficient by several standard deviations, which suggests non-perturbative contributions persist well above the QCD crossover; extending the same computation to 300–500 GeV would show whether the entropy approaches the free-gas limit with the perturbative slope.
Because the lines of constant physics are fixed by a finite-volume coupling, the method adapts directly to $N_f = 4$ or $5$ QCD; a full Standard Model equation of state from a few GeV to the electroweak scale could reduce early-Universe computations below the current few-percent systematic uncertainty.
The one-loop improvement of the entropy observable relies on lattice perturbation theory; a future comparison of $s/T^3$ obtained from energy-momentum tensor one-point functions in the same shifted-boundary setup would test this source of systematic error at the percent level.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the good news: this is a real first. Nobody has had non-perturbative results for the QCD equation of state between 3 and 165 GeV, and this group has them, with 0.5–1% errors on s/T^3. The method—shifted boundary conditions plus Schrödinger-functional coupling for lines of constant physics—is clever and avoids the usual zero-temperature subtraction. The data are extensive: four lattice spacings, one-loop improvement, explicit integration over mass and coupling with cross-checks, and the final numbers look sensible against perturbation theory. If I needed the QCD EoS for cosmology or heavy-ion work, I'd use these values with some confidence.

The soft spot is the continuum extrapolation. The final central values come from a fit with d3=0 on L0/a=6,8,10, while the O(a^3) estimate from the full dataset (including L0/a=4) is added to the errors in quadrature. That means the central values assume no a^3 artifact; if a real a^3 effect exists, it biases the intercepts. The size of that bias is estimated from the coarsest lattice, which the same analysis argues is contaminated by higher-order effects, so the error could be wrong. The paper's checks—adding a g^4 term, replacing a^3 with a^4, varying the logarithmic exponent—are reassuring and suggest the effect is small, but they don't directly test the a^3 model. Given the quoted precision is 0.5–1%, a few tenths of a percent shift is material to the central claim.

This is a proceedings contribution, and the full detail is in the companion paper (2501.11603). The reader's conditional verdict is fair: the numbers are probably right, but the error budget deserves scrutiny before treating the 0.5–1% as final. The authors are experienced and the methodology is sound; this is not overclaiming the method, just a standard extrapolation judgment call that a referee should probe.

Who should read this: lattice QCD people, cosmologists, and anyone parametrizing the SM EoS. It deserves a serious referee, and I'd send it to review with a request that the continuum extrapolation be presented in a way that doesn't fold the a^3 uncertainty into the error alone—for instance, showing how the central value shifts if a^3 is included in the fit with a prior.

Referee Report

2 major / 3 minor

Summary. The paper reports a non-perturbative lattice determination of the entropy density s/T^3 in N_f=3 massless QCD for nine temperatures between 3 GeV and 165 GeV, using shifted boundary conditions to extract s directly from the derivative of the free energy with respect to the shift. The simulations use O(a)-improved Wilson fermions, with lines of constant physics set by the non-perturbative Schrödinger functional coupling, and four lattice resolutions (L0/a=4, 6, 8, 10) for the continuum extrapolation. The final continuum values in Table 3 carry quoted relative errors of 0.5-1.0%, and are compared with high-temperature perturbation theory in Section 5. The paper is a proceedings contribution that announces and summarizes the results, referring to Ref. [14] for full details.

Significance. If correct, these are the first non-perturbative QCD equation-of-state data in the 3-165 GeV range, with a precision that would substantially improve on the poorly convergent perturbative determinations in this window. The shifted-boundary-condition method avoids the usual zero-temperature subtraction, and the paper contains several valuable internal consistency checks: the one-loop improved observable, multiple continuum fit ansätze, a logarithmic-correction scan, and a comparison with staggered results at T=500 MeV. The central claim therefore hinges on the reliability of the continuum extrapolation in Section 4.3, which is exactly the point that needs the most careful scrutiny.

major comments (2)

[§4.3, Eqs. (14)-(15)] The central values in Table 3 are obtained from a d3=0 fit to the three finest lattice spacings, while the error inflation in Eq. (15) uses d3 estimated from a d3≠0 fit to the full dataset that includes L0/a=4. This mixes two different treatments of the O(a^3) artifact: the systematic is added only to the errors, not to the central values, so a true a^3 contamination would bias the intercepts rather than being removed. The text also states that the d3=0 all-data fit gives central values 'systematically higher' than the d3≠0 fit, so the choice of central model is not neutral. In addition, the d3 estimate comes from the same L0/a=4 data that the text argues are affected by higher-order artifacts, and Eq. (15) propagates only the point estimate of d3, not its uncertainty. Please provide a d3 estimate obtained from the three finest spacings alone, an alternative with simultaneous a^3 and a^4 terms, and a quantitative statement of how the Table 3 central values and errors change under these prescriptions; without this, the quoted 0.5-1.0% accuracy is not fully supported.
[§4.3, Eq. (16)] The check of logarithmic corrections in Eq. (16) varies only the exponent γ of the a^2 term; it does not constrain the a^3 model that is the dominant systematic in Eq. (15). Since the final central values assume the a^3 term is absent, the robustness of the final extrapolation requires an analogous check for the a^3 term, for example replacing g^3 by g^{3+δ} or including log(aT) factors in the a^3 term and redetermining d3. As written, the γ∈[-1,1] scan is not sensitive to the main source of systematic uncertainty in the final result.

minor comments (3)

[§4.3, Eq. (16)] In Eq. (16) the right-hand side should read s(T_i)/T_i^3, not s(T_i)/T_i; as written the fit function is dimensionally inconsistent.
[§6, Conclusions] The statement that the final accuracy is 'dominated by the statistical error' is in tension with Eq. (15), which adds an explicit d3 systematic to every data point; please report the relative sizes of the statistical and d3 systematic contributions at each lattice spacing, or rephrase the statement.
[§4.3, general] The numerical values of d2 and d3, and the finite-lattice-spacing entropy data entering the continuum fits, are not reported in the proceedings text; a table, or an explicit reference to the corresponding table in Ref. [14], would allow the reader to assess the extrapolation more directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central entropy-density values are Monte Carlo measurements extrapolated to the continuum, with external perturbative benchmarks.

full rationale

The derivation chain is self-contained. The entropy density is obtained by applying the shifted-boundary identity of Eq. (3), from Ref. [15], to Monte Carlo measurements of the discrete shift derivative of the free energy via Eqs. (8)-(12). The continuum values in Table 3 are fit parameters of Eq. (14), i.e., extrapolated data, not quantities whose value is enforced by the fit ansatz or by the fitted parameters. The treatment of O(a^3) artifacts is a systematic-error procedure: d3 is estimated from the full dataset and added in quadrature to the errors in Eq. (15), while the central values come from the d3=0 fit on the finer lattice spacings; this may affect the reliability of the quoted errors, but it is not a circular reduction because the central result is not defined in terms of the fitted d3. The fits in Section 5 are explicitly parametrizations of the same continuum data, used for comparison with perturbation theory, and are not presented as independent predictions. The perturbative predictions themselves are external benchmarks. Self-citations [11,12,14,15] provide the method and scale-setting inputs, but the identity in Eq. (3) is a general parameter-free result and the SF coupling / Lambda inputs from [17-19] are independent non-perturbative results not fitted to the entropy density. No fitted parameter is renamed as a prediction, and no target quantity is defined in terms of the inputs. Hence no circular step is identifiable.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central entropy values are direct measurements; the only fitted numbers are continuum-fit coefficients and the Section 5 parametrizations. No new physical entities are postulated.

free parameters (3)

d2, d3 continuum fit coefficients = not quoted
Fit parameters in Eq. (14) that describe O(a^2) and O(a^3) lattice artifacts; d3 is used to define the systematic error in Eq. (15). These are fitted to the same lattice data and do not enter the observable definition.
s0, s2, s3 in Eq. (18) = s0=2.954(15), s2=-3.6(7); with SB fixed: s2=-5.1(9), s3=5(5)
Fit parameters of the polynomial parametrization of the continuum entropy data; not inputs to the entropy measurement.
qc, s7 in Eq. (19) = qc=-5.1(1.7)e3, s7=1.3(7)e4; with T=500 MeV data: qc=-4.0(1.1)e3, s7=7(4)e3
Coefficients of unknown O(g^6) and O(g^7) terms fitted to the same continuum data plus one literature point; a parametrization, not a source of the central result.

assumptions (6)

domain assumption Shifted boundary conditions define a thermal ensemble at temperature T = 1/(L0 sqrt(1+xi^2)) and physical volume L^3/sqrt(1+xi^2).
Used to set up the moving-frame path integral in Section 2, following Refs. [15,16].
standard math The entropy formula Eq. (3) follows from Euclidean Lorentz invariance of continuum QCD.
This is the identity s/T^3 = (1+xi^2)/xi_k * (1/T^4) df/dxi_k, derived in [15].
standard math The decomposition of the free-energy shift derivative into the mass integral (Eq. 11) and the static-limit pure-gauge integral (Eq. 12) is exact.
The paper rewrites f_xi minus f_infinity_xi as an integral over the subtracted quark mass; assumes the integrand is integrable and the static limit decouples quarks.
domain assumption One-loop lattice perturbation theory coefficients s0(L0/a), s2(L0/a) in Eq. (13) remove the first cutoff effects to O(g^2).
The one-loop improvement of the lattice entropy relies on standard lattice perturbation theory in the thermodynamic limit; residual effects are assumed to start at O(g^3 a^2).
domain assumption Lines of constant physics are fixed by requiring the non-perturbative SF coupling to match its continuum running at mu = T sqrt(2), Eq. (7).
This connects bare parameters to physical temperature using the ALPHA collaboration's running coupling and Lambda_MS = 341 MeV [17-19].
ad hoc to paper The continuum extrapolation ansatz Eq. (14) with a^2 g^3 and a^3 g^3 terms, and the treatment of L0/a=4 as a systematic-only estimate, describes the lattice artifacts.
The final fit excludes the coarsest spacing; the paper checks variations, but this model choice is specific to this analysis.

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Pith. "Pith review of The Equation of State of QCD up to very high temperatures." pith.science (2026). https://pith.science/paper/HYJVQ5RC

@misc{pith2026250204239,
  author       = {Pith},
  title        = {Pith review of: The Equation of State of QCD up to very high temperatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HYJVQ5RC}},
  note         = {Machine review of arXiv:2502.04239}
}

abstract

We present the non-perturbative computation of the entropy density in QCD for temperatures ranging from 3 GeV up to the electro-weak scale, using $N_f=3$ flavours of massless O$(a)$-improved Wilson fermions. We adopt a new strategy designed to be computationally efficient and based on formulating thermal QCD in a moving reference frame, where the fields satisfy shifted boundary conditions in the temporal direction and periodic boundary conditions along the spatial ones. In this setup the entropy density can be computed as the derivative of the free-energy density with respect to the shift parameter. For each physical temperature, we perform Monte Carlo simulations at four values of the lattice spacing in order to extrapolate the numerical data of the entropy density to the continuum limit. We achieve a final accuracy of approximatively $0.5$-$1.0\%$ and our results are compared with predictions from high-temperature perturbation theory.

Figures

Figures reproduced from arXiv: 2502.04239 by the authors.

**Figure 1.** Left: Derivative in the shift of the chiral condensate as a function of 𝑚𝑞/𝑇 at some selected bare parameters. Points have been interpolated with a cubic spline to guide the eye. Error bars are smaller than the markers. Right: Derivative in the shift of the pure gauge action as a function of 𝑔 2 0 for 𝐿0/𝑎 = 6. For convenience we subtracted from the data the result at one-loop order in lattice perturbation theory. t… view at source ↗

**Figure 2.** Left: The black points are the values of 𝑠/𝑇 3 at fixed lattice spacing 𝑎/𝐿0 and given temperature. Points at temperature 𝑇𝑛 have been shifted downward by 𝑛 for better readability. The red bands represent our best fit to the continuum limit. Right: Effect of logarithmic corrections on the continuum extrapolated values of the best fit, see main text for details. This leads to a total of 74 independent simulations for… view at source ↗

**Figure 3.** Left: Normalized entropy density 𝑠/𝑇 3 in the continuum limit as a function of 𝑔ˆ 2 (𝑇). The shadowed curves represent the perturbative value up to the indicated order, and the prediction from hardthermal-loop (htl) [9, 10]. The orange and blue bands (grey when the two overlap) correspond to the fits to Eqs. (18) and (19), see main text for the details. Right: 𝑠/𝑇 3 as a function of 𝑔ˆ 2 (𝑇). The black points are o… view at source ↗

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