arxiv: 2601.14967 · v2 · submitted 2026-01-21 · ✦ hep-lat · hep-ph· nucl-th

Recognition: 2 theorem links

· Lean Theorem

Shear and bulk viscosities of the gluon plasma across the transition temperature from lattice QCD

Heng-Tong Ding , Hai-Tao Shu , Cheng Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-16 12:17 UTC · model grok-4.3

classification ✦ hep-lat hep-phnucl-th

keywords lattice QCDgluon plasmashear viscositybulk viscosityspectral functionQCD transition temperaturetransport coefficients

0 comments

The pith

Gluon plasma shear viscosity to entropy ratio reaches a minimum near the transition temperature and then rises while bulk viscosity ratio falls steadily

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

Lattice QCD calculations compute the shear and bulk viscosities of pure gluon plasma from 0.76 Tc to 2.25 Tc using continuum-extrapolated energy-momentum tensor correlators. The spectral functions fitted to these correlators match perturbative ultraviolet behavior and employ Lorentzian transport peaks in the infrared, with the peak width varied over a thermal-scale range to control the main modeling uncertainty. The extracted ratios show η/s developing a minimum close to Tc before increasing in the deconfined phase, while ζ/s decreases monotonically across the entire interval. These transport coefficients supply direct input for hydrodynamic modeling of the gluon plasma near the QCD transition.

Core claim

In the gluon plasma the shear-viscosity-to-entropy-density ratio η/s exhibits a minimum near the transition temperature Tc and increases for T > Tc, whereas the bulk-viscosity-to-entropy-density ratio ζ/s decreases monotonically over the full temperature range from 0.76 Tc to 2.25 Tc.

What carries the argument

Continuum-extrapolated Euclidean correlators of the energy-momentum tensor inverted through spectral functions whose infrared part is a Lorentzian transport peak whose width is varied over a physically motivated thermal-scale range.

If this is right

Hydrodynamic simulations of gluon plasma evolution will use a lower shear viscosity near Tc than at higher temperatures.
Bulk-viscosity contributions to the stress tensor become negligible once the system moves well above the transition.
The location of the η/s minimum supplies a concrete temperature scale for the strongest dissipative effects in the deconfined phase.

Where Pith is reading between the lines

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

Including dynamical quarks may shift the minimum in η/s or alter the rate at which ζ/s falls.
The same fitting procedure applied to finer lattices or different gauge actions can test whether the minimum remains at Tc.
These viscosity curves can be used as boundary conditions for effective hydrodynamic or kinetic-theory models that bridge the transition region.

Load-bearing premise

The infrared spectral function is adequately described by a Lorentzian transport peak whose width can be bracketed by varying it over a range set by thermal scales.

What would settle it

A more precise lattice calculation or direct observation showing that η/s does not increase above Tc or that ζ/s increases anywhere in the range 0.76–2.25 Tc would contradict the reported temperature dependence.

Figures

Figures reproduced from arXiv: 2601.14967 by Cheng Zhang, Hai-Tao Shu, Heng-Tong Ding.

**Figure 2.** Figure 2: FIG. 2. Continuum extrapolation of the bare EMT correlator [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Left: flow-time extrapolation for the shear-channel correlators at [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Left: shear-channel correlators after the double extrapolation for all temperatures. Right: same as the left panel but [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The dependence of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Top left: fitted spectral functions at 0.76 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Comparison of the shear viscosity (left) and bulk viscosity (right) normalized by [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: shows that, for T < Tc, both η/s and ζ/s obtained in this work are significantly larger than the results from the analytic fit [9], the QPM [14], and the lattice determinations of Astrakhantsev et al. [10, 13]. However, when the viscosities are expressed in units of T 3 , the differences between the lattice results of Refs. [10, 13] and this work are substantially reduced, as shown in [PITH_FULL_IMAGE:f… view at source ↗

**Figure 9.** Figure 9: FIG. 9. Left: comparison of the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Top left: comparison of the continuum-extrapolated correlators from the joint and separate fits in the shear channel [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

read the original abstract

We investigate the temperature dependence of the shear viscosity ($\eta$) and bulk viscosity ($\zeta$) of the gluon plasma using lattice QCD over the range 0.76--2.25$\,T_c$, extending from below the transition temperature $T_c$ across the transition region and into the deconfined phase. At each temperature, we employ three large, fine lattices, which enables controlled continuum extrapolations of the energy-momentum tensor correlators. Using gradient flow together with a recently developed blocking technique, we achieve percent-level precision for these correlators, providing strong constraints for a model-based spectral analysis. Since the inversion to real-time information is intrinsically ill posed, we extract viscosities by fitting spectral functions whose ultraviolet behavior is matched to the best available perturbative result, while the infrared region is described by a Lorentzian transport peak. The dominant modeling uncertainty associated with the transport peak width is bracketed by varying it over a physically motivated range set by thermal scales. We find that the shear-viscosity-to-entropy-density ratio, $\eta/s$, exhibits a minimum near the transition temperature $T_c$ and increases for $T>T_c$, whereas the bulk-viscosity-to-entropy-density ratio, $\zeta/s$, decreases monotonically over the entire temperature range studied.

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

This paper gives continuum-extrapolated lattice results for gluon plasma viscosities from below Tc to 2.25 Tc, reporting a minimum in η/s near the transition, but the numbers rest on a Lorentzian model for the low-frequency spectral function.

read the letter

The main thing to know is that the authors have pushed lattice calculations of shear and bulk viscosities for the pure gluon plasma down to 0.76 Tc and through the transition, using three lattice spacings at each temperature for a controlled continuum extrapolation. They report that η/s reaches a minimum near Tc and then increases, while ζ/s falls steadily over the whole range. The technical execution on the lattice side is the clearest strength: gradient flow plus the blocking technique delivers percent-level precision on the energy-momentum tensor correlators, and the multiple spacings let them extrapolate discretization effects away in a straightforward way. That level of control on the Euclidean data is better than most earlier work in this area, which often stayed above Tc or used coarser lattices. They also anchor the ultraviolet tail of the spectral function to perturbative results, which is a reasonable step. The extraction itself uses a model spectral function with a Lorentzian transport peak in the infrared, and they vary the peak width over a thermal-scale window to estimate the dominant uncertainty. This brackets the modeling error and produces the quoted temperature trends. The soft spot is exactly where the stress-test note flags: below and near Tc the system is confined and strongly coupled, so the true low-frequency spectral shape may not be a simple Lorentzian. Thresholds, power-law tails, or extra scales could be present, and any mismatch in the infrared would shift the zero-frequency slope and therefore move or remove the reported minimum in η/s. The paper treats the width variation as the main systematic, but it is still a single-parameter ansatz. This work is aimed at people who need temperature-dependent transport coefficients for hydrodynamic modeling of heavy-ion collisions. The lattice methodology is solid enough that the paper deserves a serious referee rather than a desk reject; the model dependence can be discussed in review.

Referee Report

2 major / 2 minor

Summary. The paper computes shear (η) and bulk (ζ) viscosities of the gluon plasma on the lattice from 0.76 Tc to 2.25 Tc using continuum-extrapolated energy-momentum tensor correlators obtained with gradient flow and a blocking technique. Viscosities are extracted via a model spectral function whose UV tail is fixed to perturbative results and whose IR region is parametrized by a single Lorentzian transport peak; the dominant uncertainty is bracketed by varying the peak width over a thermal-scale range. The central results are that η/s exhibits a minimum near Tc and rises above Tc, while ζ/s decreases monotonically throughout the interval.

Significance. If the results hold, the work supplies the first controlled lattice determination of both viscosities across the deconfinement transition in pure Yang-Mills theory, with percent-level correlator precision and explicit continuum extrapolation. These temperature dependences would furnish useful benchmarks for hydrodynamic modeling of heavy-ion collisions and for testing the validity of quasiparticle or effective-theory descriptions near Tc.

major comments (2)

[Spectral-function modeling section] Spectral-function modeling section: the infrared region is described exclusively by a single Lorentzian transport peak whose width Γ is varied only over a thermal-scale window. Because η/s and ζ/s are read directly from the zero-frequency slope of this peak, any systematic deviation from Lorentzian shape (e.g., thresholds or power-law tails below Tc) propagates directly into the reported minimum of η/s and the monotonicity of ζ/s; the manuscript does not test alternative IR forms to quantify this risk.
[Results and discussion] Results and discussion: the location of the minimum in η/s is stated to lie near Tc, yet the quantitative shift of this minimum under the full range of allowed Γ values is not tabulated or plotted; without this, it is impossible to judge whether the minimum is robust or an artifact of the chosen width interval.

minor comments (2)

[Abstract] The abstract quotes the temperature range as 0.76--2.25 Tc; the main text should explicitly list the discrete temperatures simulated and confirm that the same set is used for both shear and bulk channels.
[Notation] Notation for the transport peak width (Γ) should be introduced once and used consistently; occasional switches to “width parameter” or “damping” obscure the single-parameter variation that brackets the uncertainty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to strengthen the presentation of our results.

read point-by-point responses

Referee: [Spectral-function modeling section] Spectral-function modeling section: the infrared region is described exclusively by a single Lorentzian transport peak whose width Γ is varied only over a thermal-scale window. Because η/s and ζ/s are read directly from the zero-frequency slope of this peak, any systematic deviation from Lorentzian shape (e.g., thresholds or power-law tails below Tc) propagates directly into the reported minimum of η/s and the monotonicity of ζ/s; the manuscript does not test alternative IR forms to quantify this risk.

Authors: We acknowledge that testing alternative infrared parametrizations would further quantify modeling uncertainties. The Lorentzian form is motivated by hydrodynamic expectations for the transport peak and is consistent with prior lattice QCD studies of viscosities. To address the referee's concern, we will add in the revised manuscript a supplementary analysis using an alternative IR form (e.g., Lorentzian plus a simple power-law tail) for representative temperatures below and above Tc. This will be included in an expanded spectral-function modeling section to demonstrate robustness of the reported trends. revision: yes
Referee: [Results and discussion] Results and discussion: the location of the minimum in η/s is stated to lie near Tc, yet the quantitative shift of this minimum under the full range of allowed Γ values is not tabulated or plotted; without this, it is impossible to judge whether the minimum is robust or an artifact of the chosen width interval.

Authors: We agree that explicitly documenting the dependence of the minimum location on Γ would improve clarity. In the revised manuscript we will add a new table (and accompanying plot) in the Results and discussion section that reports the position of the η/s minimum for each value of Γ in the allowed thermal-scale range. This will show that the minimum remains near Tc across the variations and is not an artifact of the specific width interval chosen. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extraction is data-driven with explicit model assumptions

full rationale

The paper computes Euclidean correlators on the lattice with gradient flow and blocking, then fits a model spectral function (UV matched to perturbation theory, IR Lorentzian transport peak) to extract viscosities at each temperature. The temperature dependence of η/s and ζ/s follows directly from independent fits at each T, with the width parameter varied over a thermal-scale range to bracket uncertainty. No step reduces the reported minimum in η/s or the monotonic decrease in ζ/s to the input data by construction, nor is any uniqueness theorem or ansatz smuggled via self-citation in a load-bearing way. The 'recently developed blocking technique' improves precision but is not required to justify the qualitative claims. The derivation remains self-contained against the lattice data.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the low-frequency spectral function is well-described by a single Lorentzian whose width can be varied over thermal scales, plus the matching of the UV tail to perturbative QCD; these introduce the dominant systematic uncertainty.

free parameters (1)

Lorentzian transport peak width
Varied over a physically motivated range set by thermal scales to bracket the dominant modeling uncertainty in the spectral reconstruction.

axioms (2)

domain assumption The ultraviolet behavior of the spectral function matches the best available perturbative QCD result
Used to constrain the high-frequency part of the model spectral function.
ad hoc to paper The infrared region of the spectral function is described by a single Lorentzian transport peak
Assumed for the model-based extraction of viscosities from the Euclidean correlators.

pith-pipeline@v0.9.0 · 5534 in / 1556 out tokens · 60120 ms · 2026-05-16T12:17:04.309694+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the infrared region is described by a Lorentzian transport peak whose width is varied over a physically motivated range set by thermal scales
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We find that the shear-viscosity-to-entropy-density ratio, η/s, exhibits a minimum near the transition temperature Tc

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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