arxiv: 2604.19649 · v1 · submitted 2026-04-21 · ✦ hep-lat · hep-ph· hep-th

Recognition: unknown

Finite-density equation of state of hot QCD using the complex Langevin equation

Michael Mandl , D\'enes Sexty , Daniel Unterhuber

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:30 UTC · model grok-4.3

classification ✦ hep-lat hep-phhep-th

keywords QCD equation of statefinite baryon densitycomplex Langevinlattice QCDbaryon chemical potentialhot QCDcontinuum extrapolation

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The pith

Lattice simulations using the complex Langevin equation determine the QCD equation of state at high baryon densities.

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

The paper applies the complex Langevin equation to continuum-extrapolated lattice QCD simulations at the physical point, above the crossover temperature and at baryon densities higher than previously accessible. It computes the baryon density and pressure directly as functions of baryon chemical potential and temperature, thereby obtaining the equation of state. Convergence of the stochastic dynamics is shown to be reliable, producing results that match both earlier lattice calculations at smaller chemical potentials and perturbative hard-thermal-loop predictions at high temperatures. A sympathetic reader cares because the equation of state governs the thermodynamics of hot, dense matter created in heavy-ion collisions and present in neutron stars.

Core claim

The QCD equation of state is obtained by computing the baryon density and the pressure on the lattice as functions of baryon chemical potential and temperature, using the complex Langevin equation for continuum-extrapolated simulations at the physical point and unprecedentedly high densities, with the dynamics shown to converge correctly and to agree with known limits.

What carries the argument

The complex Langevin equation, which evolves the gauge fields in a complexified space to sample the QCD path integral with a complex fermion determinant at finite chemical potential.

If this is right

Thermodynamic quantities such as energy density and entropy density follow directly from the pressure and baryon density.
The results extend the range of reliable lattice data for use in hydrodynamic modeling of heavy-ion collisions.
Consistency with hard-thermal-loop perturbation theory at high temperature supports the method in the deconfined regime.
The controlled convergence allows systematic studies at densities where the sign problem blocks standard Monte Carlo methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same framework could be used to compute higher-order susceptibilities that probe the vicinity of a possible critical endpoint.
The equation of state tables produced here could be directly imported into simulations of neutron-star mergers.
Extension of the method to lower temperatures might eventually map the full phase diagram if convergence remains under control.

Load-bearing premise

The complex Langevin dynamics converges to the correct physical distribution rather than a spurious solution.

What would settle it

A direct comparison at overlapping values of temperature and chemical potential showing disagreement with an independent method, such as Taylor expansion results from other groups or resummed perturbation theory at high temperature, would falsify the reliability of the computed equation of state.

Figures

Figures reproduced from arXiv: 2604.19649 by Daniel Unterhuber, D\'enes Sexty, Michael Mandl.

**Figure 3.** Figure 3: Once more one observes reasonable agreement between the results for small µB/T and larger deviations upon increasing the chemical potential. Rather consistently, the HTL results lie between the free theory and our nonperturbative results. Lines of constant baryon density nB (normalized by the nuclear density n0 = 0.16 fm−3 ) in the (µB/T, T) plane are shown in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Lines of constant baryon density [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Pressure difference (normalized by [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

read the original abstract

We present the results of continuum-extrapolated lattice simulations of quantum chromodynamics (QCD) above the crossover temperature and for unprecedentedly high baryon densities at the physical point, employing the complex Langevin equation. In particular, we determine the QCD equation of state by computing the baryon density as well as the pressure as functions of the baryon chemical potential and the temperature. Potential issues with wrong convergence of complex Langevin dynamics are under control and we indeed find agreement with previous lattice studies working at smaller chemical potentials, as well as with perturbative hard-thermal-loop calculations at high temperatures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

CLE reaches continuum-extrapolated physical-point QCD EOS at higher baryon densities than before, with the usual convergence question still central.

read the letter

This paper runs complex Langevin simulations of QCD above the crossover, at physical quark masses, and extracts the equation of state out to higher baryon chemical potentials than earlier lattice work, with a continuum extrapolation included. They report the baryon density and pressure versus mu_B and T, and state that the results line up with previous lattice data at smaller mu and with hard-thermal-loop perturbation theory at high T. That reach is the concrete step forward; the method itself is not new, but combining it with full extrapolation at the physical point and pushing the density range is incremental progress worth noting. The abstract is short on the exact diagnostics, but the claim is that wrong-convergence problems are under control. The stress-test note is right that this is the load-bearing piece: if the drift norms, imaginary-action distributions, or other checks hold quantitatively at the largest mu values, the high-density part stands; if they degrade, the extrapolation becomes unreliable while the low-density agreement can still look fine. The paper does not appear to invent new entities or hide free parameters, and it is a direct simulation result rather than a fit. For people who need first-principles EOS input in the high-density regime for heavy-ion or neutron-star modeling, this supplies usable numbers if the convergence holds. The thinking is straightforward and engages the existing literature on CLE limitations. I would bring it to a reading group to go through the diagnostics in detail. It is not something I would cite in my own work in the next year, but a serious editor should send it to referees rather than desk-reject it; the numerical advance is real enough to merit external scrutiny even if revisions on the convergence section are likely.

Referee Report

2 major / 1 minor

Summary. The manuscript reports continuum-extrapolated lattice QCD simulations above the crossover temperature at the physical point, using the complex Langevin equation to compute the baryon density and pressure as functions of baryon chemical potential μ_B and temperature T. It reaches unprecedentedly high densities and asserts that wrong-convergence issues of the complex Langevin dynamics are under control, with results agreeing with prior lattice work at smaller μ_B and with hard-thermal-loop perturbation theory at high T.

Significance. If the convergence control is rigorously established, the work would provide a significant extension of lattice QCD into the high-density regime inaccessible to standard Monte Carlo methods due to the sign problem. The direct simulation yields the EOS without fitted parameters or self-referential definitions, offering falsifiable predictions at physical quark masses and continuum limit that can be compared to heavy-ion phenomenology and other non-perturbative approaches.

major comments (2)

[Abstract] Abstract: the central claim that 'potential issues with wrong convergence of complex Langevin dynamics are under control' is load-bearing for the entire EOS determination at high μ_B, yet the abstract (and by extension the manuscript) provides no explicit quantitative diagnostics, thresholds, or figures showing the distribution of the imaginary part of the action, drift-term norms, or consistency checks with reweighting at the largest accessed densities.
[Results] Results section: the stated agreement with previous lattice studies at smaller chemical potentials and with HTL perturbation theory lacks reported error budgets, quantitative deviation measures, or details of the continuum-extrapolation procedure (e.g., fit forms, number of lattice spacings, or χ² values), preventing assessment of whether the high-density extrapolation is reliable or merely consistent within large uncertainties.

minor comments (1)

[Abstract] Abstract: the phrase 'unprecedentedly high baryon densities' should be accompanied by a concrete range or maximum value of μ_B/T to allow immediate gauging of the advance relative to prior work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each point below and will incorporate revisions to improve the clarity and completeness of the presentation.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that 'potential issues with wrong convergence of complex Langevin dynamics are under control' is load-bearing for the entire EOS determination at high μ_B, yet the abstract (and by extension the manuscript) provides no explicit quantitative diagnostics, thresholds, or figures showing the distribution of the imaginary part of the action, drift-term norms, or consistency checks with reweighting at the largest accessed densities.

Authors: We agree that the abstract would benefit from a more explicit reference to the convergence diagnostics. While the main text (particularly the sections on simulation setup and validation) already presents quantitative checks—including histograms of the imaginary part of the action, norms of the drift term, and comparisons to reweighting at moderate densities—we will revise the abstract to briefly mention these controls and their outcomes at the highest μ_B values. This will make the load-bearing claim more self-contained without altering the manuscript's technical content. revision: yes
Referee: [Results] Results section: the stated agreement with previous lattice studies at smaller chemical potentials and with HTL perturbation theory lacks reported error budgets, quantitative deviation measures, or details of the continuum-extrapolation procedure (e.g., fit forms, number of lattice spacings, or χ² values), preventing assessment of whether the high-density extrapolation is reliable or merely consistent within large uncertainties.

Authors: We acknowledge that additional quantitative details will aid assessment. The manuscript already performs a continuum extrapolation using multiple lattice spacings and reports statistical errors, but we will expand the results section to include: (i) explicit error budgets separating statistical and systematic contributions for the EOS quantities, (ii) quantitative deviation measures (e.g., relative differences and significance in units of combined uncertainty) for the comparisons to prior lattice data and HTL, and (iii) full details of the extrapolation procedure, including the fit functional form, the number of spacings employed, and the resulting χ² values. These additions will be presented in a revised results section and associated figures/tables. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical lattice computation of the EOS via complex Langevin

full rationale

The paper reports continuum-extrapolated lattice results for the baryon density and pressure obtained by direct stochastic sampling with the complex Langevin equation. Thermodynamic relations are used to obtain the pressure from the density (standard integration dP = n_B dμ_B), but this is an exact identity applied to independently computed observables rather than a self-referential fit or redefinition. Validation against lower-μ lattice data and high-T perturbation theory is presented as consistency check, not as load-bearing input. No derivation step reduces by construction to fitted parameters, self-citations, or ansätze imported from prior work by the same authors; the central output is the numerical evaluation itself under stated convergence diagnostics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The simulation rests on standard lattice QCD regularization and the assumption that complex Langevin correctly samples the complex measure without uncontrolled bias.

axioms (2)

domain assumption Complex Langevin dynamics converges to the correct complex measure for the QCD action at finite density
Invoked to justify the method's validity; stated as under control in the abstract.
standard math Continuum limit exists and can be reached by extrapolation from finite lattice spacings
Standard assumption in lattice field theory.

pith-pipeline@v0.9.0 · 5398 in / 1144 out tokens · 68672 ms · 2026-05-10T00:30:10.966269+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Lattice field theories with a sign problem
hep-lat 2026-04 unverdicted novelty 2.0

A review of holomorphic extensions, dual variables, tensor renormalization group, and machine learning approaches for controlling the sign problem in lattice field theories.
Lattice field theories with a sign problem
hep-lat 2026-04 unverdicted novelty 1.0

Reviews approaches such as Lefschetz thimbles, complex Langevin dynamics, dual variables, tensor renormalization group, and machine learning to control the sign problem in lattice field theories.

Reference graph

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within the region of the phase diagram that is ac- cessible to both studies. In addition, for smallµ B/Twe find agreement with hard-thermal-loop (HTL) perturba- tive results derived in [59–61], as well as with the baryon density of a theory of noninteracting quarks and gluons, only the former of which contribute ton B. It should be noted that we have naiv...
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and find no disagreement. Furthermore, as before 4 0 2 4 6 8 10 12 14 16 µB/T 0 10 20 30 40 50 60p/T 4 Free theory T = 600 MeV HTL T = 600 MeV T = 600 MeV T = 360 MeV T = 275 MeV T = 260 MeV Saturation eﬀects 0 1 2 3 4 5 3 4 5 6 7 8 Bors´ anyi et al. T = 275 MeV Bors´ anyi et al. T = 260 MeV FIG. 2: Similar as Fig. 1, but for the pressure. we provide comp...

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in our entire temperature range using (10) and (11). In [60], HTL results for the pressure difference ∆pas a function ofTwere compared with (older) lattice data for different small chemical potentials and good agree- ment was found. A similar comparison of our results for ∆pwith HTL perturbation theory and a free-theory cal- culation but for larger chemic...
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Further inter- esting research directions include the addition of electro- magnetic fields [71, 72] and/or an isospin chemical po- tential [73]

of machine-learning so-called kernels that could sta- bilize simulations in the low-temperature regime of QCD like they have in other systems [66–70]. Further inter- esting research directions include the addition of electro- magnetic fields [71, 72] and/or an isospin chemical po- tential [73]. We would like to thank Szabolcs Bors´ anyi and Jana G¨ unther...

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