Dense QC$_2$D. What's up with that?!?

Andrew Lee-Mitchell; Dale Lawlor; Jon-Ivar Skullerud; Seyong Kim; Simon Hands

REVIEW 3 major objections 5 minor 1 cited by

Dense QC$_2$D. What's up with that?!?

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In dense two-color QCD, the squared speed of sound exceeds the conformal limit 1/3.

desk verdict A plausible result that overreaches on error control: the ∂κ/∂a uncertainty is a load-bearing gap, so 'clearly breaches' is not yet backed. read the letter →

arxiv 2412.15872 v1 pith:ZJTH2STO submitted 2024-12-20 hep-lat hep-phnucl-th

Simon Hands , Seyong Kim , Dale Lawlor , Andrew Lee-Mitchell , Jon-Ivar Skullerud This is my paper · ORCID

classification hep-lathep-phnucl-th PACS 11.15.Ha12.38.Gc12.38.Mh

keywords two-colorQCDlatticespeedofsoundequationstateconformallimitdiquarksuperfluidfinite-densitysimulationstaticquarkpotential

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 paper reports lattice simulations of two-color QCD (QC2D) at non-zero baryon density, a sign-problem-free testbed for dense QCD matter. The authors compute the equation of state on a finer lattice than earlier work, with a new scale-setting procedure, and extract the speed of sound. They find that the squared speed of sound $C_s^2$ rises sharply after the onset chemical potential and clearly exceeds the conformal limit $1/3$ over a range of densities. Because the result is consistent with earlier coarser-lattice determinations and with isospin QCD simulations, it supports the idea that the breach of the conformal bound is a robust property of dense strongly interacting matter rather than a lattice artefact.

What carries the argument

The central object is the squared speed of sound $C_s^2 = dP/d\varepsilon$, built from the pressure $P$ and the energy density $\varepsilon = T^\mu_\mu + 3P$, where the trace anomaly $T^\mu_\mu$ is split into gluonic and fermionic parts renormalised by the $\beta$ functions $\partial\beta/\partial a$ and $\partial\kappa/\partial a$. The paper's new ingredient is a scale-setting chain that fixes the lattice spacing from the Cornell form of the static quark potential, using Coulomb-gauge Wilson lines rather than Wilson loops, and determines the $\beta$ functions from fits to $\beta(a)$ and $\kappa(a)$ along the line of constant physics. The thermodynamic observables are extrapolated to zero diquark source $j$ after simulations with $aj = 0.01$--$0.03$ that lift the low-lying modes of the superfluid phase. The conformal limit $C_s^2 = 1/3$ serves as the benchmark against which the dense-matter equation of state is compared.

What would settle it

A calculation of the same speed-of-sound curve on a second, finer lattice spacing on the same line of constant physics, extrapolated to the continuum at fixed $\mu_q/m_\pi$, would settle the claim: if the continuum value does not exceed $1/3$ for $\mu_q/m_\pi$ around 1.0--1.4, the reported breach is a lattice artefact.

Watch

Extended reading notes

Core claim

Using unimproved Wilson fermions and gauge action on a lattice with $\beta=2.1$, $\kappa=0.1577$, and spacing $a=0.130$ fm set by the Cornell static-quark potential, the authors compute the equation of state of dense two-color QCD at quark masses on the line of constant physics $m_\pi/m_\rho \simeq 0.81$. The pressure is obtained by integrating the quark number density in $\mu_q$, with lattice artefacts mitigated by a Stefan--Boltzmann quotient (Scheme II), and the energy density follows from the trace anomaly using $\beta$ functions determined from the new scale setting. The central result, stated in Section 3.5, is that $C_s^2 = dP/d\varepsilon$ rises sharply above the onset chemical potential $\mu_0 = m_\pi/2$ and clearly breaches the conformal limit $1/3$, before later falling. The authors report that this behaviour is consistent with other recent QC2D calculations and with isospin QCD simulations, while remaining below the upper bound derived from relativistic hydrodynamics.

Load-bearing premise

The physical scale is set by assuming the Cornell static-quark potential with string tension $\sigma=(440\text{ MeV})^2$ holds for two-color QCD, and that Coulomb-gauge Wilson lines match Wilson-loop results within $2\sigma$; if either assumption fails, the $\mu$ axis of the speed-of-sound curve shifts and the comparison with the conformal bound could change.

Editorial extensions

If this is right

If the lattice result survives the continuum limit, the speed of sound in dense two-color QCD exceeds $1/3$ for a range of chemical potentials above onset, meaning the conformal bound is not a universal property of QCD-like matter.
The consistency of the finer-lattice result with earlier coarser-lattice results [8-10] supports the conclusion that the excess over $1/3$ is not a lattice-spacing artefact.
The new scale-setting procedure based on the Cornell potential gives beta functions consistent with Karsch coefficients on the coarse lattice, validating the fine-lattice thermodynamics.
The computed $C_s^2$ remains below the upper bound derived from relativistic hydrodynamics [22], so the two constraints on the speed of sound are compatible.
The diquark condensate extrapolated to $j=0$ is non-zero above onset and the chiral perturbation theory exponent $C_2 = 1/3$ fails at high densities, indicating that the superfluid phase is not $\chi$PT-dominated there.

Reading between the lines

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

If the same breach of the conformal bound persists in the continuum limit, the equation of state of dense strongly interacting matter would be stiffer than conformal in a window above nuclear saturation, with potential consequences for neutron-star radii inferred from gravitational-wave and X-ray observations.
The Wilson-line method for extracting the static potential, once cross-checked against Wilson loops, could reduce the cost of scale setting on finer lattices and make continuum extrapolations of dense-matter thermodynamics more tractable.
The observed failure of $\chi$PT at high density suggests the superfluid may cross over to a BEC-BCS-type regime; this could be tested on the same ensembles by measuring the diquark pair radius or the fermion dispersion relation.
Applying the same thermodynamic analysis to isospin-asymmetric QCD could reveal whether the $C_s^2 > 1/3$ window near onset is a universal feature of dense QCD-like theories.

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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

Simon Hands and colleagues have put out a short, honest proceedings paper. The headline—C_s^2 exceeds the conformal bound in dense two-color QCD—is not new; Itou and Iida already reported it, and the authors say their finer-lattice data are consistent with refs [8–10]. What is new: a fine lattice at β=2.1 with a=0.130 fm, beta functions determined on a line of constant physics rather than Karsch coefficients, a Wilson-line scale-setting method, and a released analysis suite with data. That last item is real value; the code and datasets are on Zenodo.

The central calculation follows the standard thermodynamic-integration route, with a check against the Stefan-Boltzmann pressure and good consistency for the coarse-lattice beta functions. The paper is transparent about its weaknesses in Section 4: autocorrelations untreated, errors bootstrapped rather than jackknifed, fit errors read off the fitting function, low diquark-source runs at small volume. Those are all acknowledged.

The soft spot that matters more is one the authors do not flag. The fermionic trace anomaly, Eq. (13), is multiplied by a ∂κ/∂a measured as 0.152 ± 0.32 fm^{-1} on the fine lattice—relative uncertainty above 200%, consistent with zero. That uncertainty is not propagated into ε or into C_s^2. The stress-test concern holds up: if ∂κ/∂a sits at its lower end, the fermionic contribution to the trace anomaly changes sign and magnitude, and the breach above 1/3 may not survive. This is a load-bearing gap, not a cosmetic one. The same issue affects the comparison with other groups, since the same coefficient enters the scale setting.

There are two smaller reservations. The lattice spacing relies on a Cornell string tension tuned to real QCD and on a Wilson-line/Wilson-loop equivalence that is still 'in preparation' (Section 2.2). And the choice of Scheme II over Scheme I is justified mostly by matching SB at high density, which is reasonable but post hoc.

So the paper is worth reading by anyone tracking the dense-QCD EoS, and the released data will be reused. But 'clearly breaches' is not supported by the error accounting as it stands. I would not press for the claim to be withdrawn; I would press for the uncertainty in ∂κ/∂a to be propagated and for the matter to be checked on larger volume before anyone builds on the number. For a proceedings paper this level of candor is good; for a central claim it needs another pass. I'd accept this for peer review because the question is important and the group is credible, but with the expectation of revision.

Referee Report

3 major / 5 minor

Summary. This proceedings paper reports new lattice simulations of two-color QCD (QC2D) at non-zero baryon chemical potential, with the stated goal of computing the speed of sound C_s^2 = dP/dε on a finer lattice (β = 2.1, a ≃ 0.130 fm) than earlier studies. The authors tune to a line of constant physics m_π/m_ρ ≃ 0.81, extract beta functions from fits to scale-setting data, compute the pressure by integrating the quark number density with scheme-II lattice-artefact corrections, and obtain the energy density from the trace anomaly via Eqs. (2)-(3). The central result, stated in §3.5, is that C_s^2 rises sharply above the onset chemical potential and 'clearly breaches' the conformal limit 1/3. The paper also describes a new Wilson-line-based static potential method for scale setting and releases the data and analysis code on Zenodo.

Significance. If confirmed, the finding that C_s^2 exceeds the conformal bound in dense two-color QCD supports and extends similar observations in independent lattice studies and in isospin QCD, with potential implications for neutron-star equations of state. The paper's use of a finer lattice and a different scale-setting procedure than earlier works is a valuable cross-check. The explicit release of ensembles, analysis code, and speed-of-sound data (Zenodo records [23], [25], [26]) is a significant strength for reproducibility. However, the headline claim is not yet backed by a quantitative error budget: the uncertainties in the beta functions, the diquark-source extrapolation, and the derivative of the interpolated pressure-energy curve are not propagated into C_s^2. The authors themselves acknowledge in §4 that the error analysis is 'crude', so the current manuscript should be treated as a preliminary report rather than a definitive measurement.

major comments (3)

[§3.4, Eq. (13); Table 2] The fermionic trace anomaly in Eq. (13) is proportional to a (∂κ/∂a). For the fine lattice (β = 2.1) Table 2 quotes ∂κ/∂a = 0.152 ± 0.32 fm^{-1}, i.e. a relative uncertainty above 200% and a value consistent with zero. With κ^{-1} ≃ 6.34 and the subtracted chiral condensate of Fig. 5b (magnitude ~0.008 in lattice units), the 1σ uncertainty in T_q^μμ is comparable to or larger than the total trace-anomaly signal shown in Fig. 4. The manuscript does not propagate this uncertainty through Eq. (2) into ε or into C_s^2 = dP/dε. If ∂κ/∂a lies near the lower end of its error bar, the fermionic contribution changes sign and magnitude, and the breach of the conformal bound in §3.5 may not survive. This is the central load-bearing issue: the claim 'C_s^2 clearly breaches' is not supported by the current error accounting. The authors should either (a) propagate all fit and statistical uncertainties into ε and C_s^2, or (b) constrain ∂κ/∂a with additional scale-setting data, or (c) explicitly weaken the claim to a qualitative observation pending a full error budget.
[§4; §3.2] The Discussion acknowledges that measurements are taken every trajectory with possible autocorrelations, that errors are bootstrapped rather than jackknifed, and that fit errors are 'currently read off of the fitting function'. Separately, §3.2 states that the lowest diquark-source runs were performed on the smaller L_s = 16^3 spatial volume, while the highest source used N_s = 24. These are exactly the conditions that control the diquark extrapolation (Eq. (8)) used to obtain the zero-source observables that feed into the pressure and trace anomaly. Without an estimate of the systematic error from the mixed-volume diquark extrapolation, and without a confidence band on the derivative of the cubic-spline interpolation in Fig. 6b, the reported C_s^2 values in Fig. 7 have no demonstrated validity. The authors should provide the central values and uncertainties of C_s^2 as a function of μ_q, even if preliminary, so the reader can judge whether the conformal-limit breach is statistically significant.
[§2.2, Eq. (5)] The lattice spacing is set by assuming the Cornell static-quark potential with the three-color QCD string tension σ = (440 MeV)^2, and the Wilson-line method in Coulomb gauge is claimed to agree with Wilson-loop results 'within 2σ' but only as 'results in preparation'. While the dimensionless ratio C_s^2 is invariant under a global change of scale, the physical units of the chemical potential and the comparison with other groups' scale-setting choices (e.g. T = 200 MeV in refs. [8-10]) depend on this assumption. The manuscript should either provide the validation of the Wilson-line/Wilson-loop equivalence as an appendix or reference, or explicitly state the sensitivity of the speed-of-sound curve to the assumed string tension.

minor comments (5)

[§3.5] The sentence 'There is a sharp increase above at the onset chemical potential' appears to be missing a word; it should read 'above the onset chemical potential'.
[Figure 4] The y-axis notation ('−10×') is unclear; please specify the power of ten and the units of T^μμ (e.g., lattice units a^{-4} or normalized by μ_c^4).
[§3.4, Eq. (13)] The text says the μ = 0 values are subtracted to obtain Fig. 5; please clarify whether the subtraction is applied to the entire operator (4N_f N_c − ⟨ψ̄ψ⟩) or only to ⟨ψ̄ψ⟩, since the constant term may cancel in the difference.
[Figure 6b] Please specify whether the 'cubic spline interpolation' is a smoothing spline or an interpolating spline through all points, and how many knots or the smoothing parameter were used, since the speed of sound is obtained by differentiating this curve.
[Abstract] The abstract describes the paper as 'recent updates and results'; since this is a proceedings contribution, it may be helpful to state that these are preliminary results and that a full error analysis is in progress, consistent with the Discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the speed-of-sound result is derived from measured thermodynamic quantities, not from a fitted parameter renamed as a prediction.

full rationale

The central claim, C_s^2 > 1/3, is not equivalent to any input of the calculation. Equation (1) defines C_s^2 = dP/dε. The pressure P is obtained in Eq. (10) by integrating the measured quark-number density n_q with a parameter-free Stefan-Boltzmann correction, and the energy density ε is obtained from Eqs. (2), (3), (12), and (13) using measured plaquette and chiral-condensate values multiplied by the beta functions ∂β/∂a and ∂κ/∂a. The beta functions are fitted to lattice spacings along the line of constant physics, but the fitted quantities are the beta functions themselves, not the speed of sound; C_s^2 only emerges after differentiating an interpolated P(ε) curve. The conformal bound 1/3 is not a fit parameter. The paper also checks its results against independent recent QC2D works [8-10] and isospin QCD simulations, and against Stefan-Boltzmann benchmarks. The self-citations [1-7,16,23-26] provide code, prior methodology, and a standard proof of the absence of a sign problem; none is load-bearing for the headline claim, and the independent works [8-10] are by different authors. Two caveats are genuine concerns but are not circularity: the lattice spacing relies on an assumed (440 MeV)^2 string tension and on a Wilson-line/Wilson-loop equivalence that the paper states is still 'results in preparation' (Sec. 2.2), and the fermionic beta function ∂κ/∂a carries a large uncertainty (Table 2). These are calibration and error-accounting risks; they do not make the derivation reduce to its own inputs. No circular step is therefore established.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The calculation rests on standard lattice-scale-setting assumptions: a fixed string tension, the Cornell form of the static quark potential, the equivalence of Wilson lines and Wilson loops, and Stefan-Boltzmann corrections. These are not derived in the paper and are the main upstream assumptions. The diquark source and extrapolation fits add fitted parameters, but no new physical entities or forces are introduced.

free parameters (5)

Cornell potential parameters alpha_s and V0 = not reported
Eq. (5) fits Wilson-line static quark potential data; these parameters, together with the fixed string tension sigma=(440 MeV)^2, determine the lattice spacing a used for all dimensionful quantities.
beta(a) fit coefficients = quadratic coefficients from Fig. 1a
The fit is used to compute d(beta)/da = -2.85 +/- 0.10 at beta=2.1, entering the gluonic trace anomaly in Eq. (12).
kappa(a) fit coefficients = fit coefficients from Fig. 1b
The fit is used to compute d(kappa)/da = 0.152 +/- 0.32 at beta=2.1, entering the fermionic trace anomaly in Eq. (13).
diquark condensate fit parameters C0, C1, C2 = per-chemical-potential fits; C2 ~ 1/3 near onset
Eq. (8) extrapolates the diquark condensate to a_j = 0; the value C2 = 1/3 is chiPT-inspired near onset but the authors note it is violated at high density.
linear diquark extrapolation slopes for n_q, plaquette, chiral condensate = not reported
Linear fits in the diquark source are used in Sections 3.3 and 3.4 to extrapolate the quark number density, plaquette sum, and chiral condensate to a_j = 0.

assumptions (6)

standard math SU(2) gauge theory with an even number of quark flavours has no complex action problem.
Invoked in Section 1 to justify simulating QC2D at finite density; proof cited to [16].
domain assumption The string tension sigma=(440 MeV)^2 provides the physical scale for QC2D.
Section 2.2 sets the lattice spacing from the Cornell fit with this fixed string tension; all dimensionful results scale from it.
domain assumption The static quark potential is described by the Cornell form in Eq. (5).
Section 2.2 fits Wilson-line data to aV(r) = -alpha_s (N_c^2 - 1)/N_c / r + a^2 sigma r + a V0.
domain assumption Wilson lines fixed to Coulomb gauge give the same static quark potential as Wilson loops.
Section 2.2 says the two agree within 2 sigma, but the comparison is described as 'results in preparation'; the speed-of-sound analysis relies on this scale setting.
domain assumption The Stefan-Boltzmann correction Scheme II removes lattice IR and UV artefacts from the pressure.
Section 3.3 obtains pressure by integrating the quark number density and renormalizing with n_Cont/n_Lat; Scheme II is chosen because it best approximates the continuum SB result at higher density.
domain assumption The quark number density correction can be evaluated on a larger spatial volume (4 N_s) than the simulated volume.
Section 3.3 follows refs [4,7] in using 4 N_s to evaluate the lattice quark number density used in the correction factor.

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Pith. "Pith review of Dense QC$_2$D. What's up with that?!?." pith.science (2026). https://pith.science/paper/ZJTH2STO

@misc{pith2026241215872,
  author       = {Pith},
  title        = {Pith review of: Dense QC$_2$D. What's up with that?!?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJTH2STO}},
  note         = {Machine review of arXiv:2412.15872}
}

abstract

We present recent updates and results from QC$_2$D (Two Colour QCD) simulations at non-zero baryon density, including progress toward determining the speed of sound.

Figures

Figures reproduced from arXiv: 2412.15872 by the authors.

**Figure 2.** Diquark Condensate 3.1 Simulation Code The new gauge ensembles and scale setting data were produced over six weeks using the code in [23]. The major improvements over the original [1–7] code are a mixed precision conjugate gradient, changing to the RANLUX [24] generator, improved hybrid OpenMP/MPI support on CPU based machines and a CUDA port. These ensembles were generated using the CUDA version of the code. The sc… view at source ↗

**Figure 3.** Number density and pressure corrected by their SB values. be found in [7] and [1] respectively. 𝑛 Lat 𝑞 = 4𝑁𝑓 𝑁𝑐 𝑁 3 𝑠 𝑁𝑡 ∑︁ 𝑘 𝑖 sin ˜𝑘0 Í 𝑖 cos 𝑘𝑖 − 1 2𝜅 1 2𝜅 − Í 𝜈 cos ˜𝑘𝜈 2 + Í 𝜈 sin2 ˜𝑘𝜈 (9) As was discussed in [4, 7] in order to evaluate 𝑛 Lat 𝑞 one must consider a larger spatial volume than the one actually used. For this work, 4𝑁𝑠 was considered. We interpolate the quark number density using a cubic sp… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Trace Anomaly The beta functions 𝜕𝛽 𝜕𝑎 and 𝜕𝜅 𝜕𝑎 were evaluated in section 2. Thus all that is left to do is evaluate the plaquette sum and ⟨𝜓𝜓¯ ⟩ as seen in figure 5. The plaquette sum and ⟨𝜓𝜓¯ ⟩ depend weakly on the diquark source so a linear fit was again used for …

**Figure 5.** Figure 5: Zero subtracted and diquark extrapolated plaquette sum and chiral condensate. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 q aμ 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6mπ q μ 4 c μ ε j=0.000 4 c μ P (a) 𝑃 and 𝜀 normalised by the onset chemical potential 𝜇𝑐 = 𝑚𝜋. 0 2 4 6 8 10…

**Figure 6.** Figure 6: Pressure and Energy Density 7 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Speed of sound. Upper region denotes area forbidden by relativistic hydrodynamics in [22]. Whilst these early results are promising, there is still more work to do. Measurements have been taken for every trajectory, meaning there may be autocorrelations present. The lo…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Phase and equation of state of finite density QC$_2$D at lower temperature
hep-lat 2024-12 conditional novelty 2.0 of 10

Proceedings summarizing lattice QC2D results: the conformal bound c_s^2/c^2 = 1/3 is exceeded in the BCS phase at T = 40 and 80 MeV, with a rich hadronic/BEC/BCS phase structure.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Lawlor, S

D. Lawlor, S. Hands, K. Seyong and J.-I. Skullerud,su2hmc, July, 2024. 10.5281/zenodo.12910604

work page doi:10.5281/zenodo.12910604 2024

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Skullerud and D

J.-I. Skullerud and D. Lawlor,QC2D Analysis Suite, Dec., 2024. 10.5281/zenodo.14503846

work page doi:10.5281/zenodo.14503846 2024

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Lawlor and J.-I

D. Lawlor and J.-I. Skullerud,Dense QC2D. What’s up with that?!? The dataset., Nov., 2024. 10.5281/zenodo.14201453. 10

work page doi:10.5281/zenodo.14201453 2024

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Hands, S

S. Hands, S. Kim and J.-I. Skullerud,Deconfinement in dense 2-color QCD,Eur. Phys. J. C48(2006) 193 [hep-lat/0604004]

arXiv 2006

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Hands, S

S. Hands, S. Kim and J.-I. Skullerud,A Quarkyonic Phase in Dense Two Color Matter?,Phys. Rev. D 81(2010) 091502 [1001.1682]

arXiv 2010

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T. Boz, S. Cotter, L. Fister, D. Mehta and J.-I. Skullerud,Phase transitions and gluodynamics in 2-colour matter at high density, Eur. Phys. J. A49 (2013) 87 [1303.3223]

arXiv 2013

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Cotter, P

S. Cotter, P. Giudice, S. Hands and J.-I. Skullerud,Towards the phase diagram of dense two-color matter,Phys. Rev. D87(2013) 034507 [1210.4496]

arXiv 2013

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Cotter,Non-perturbative determination of Karsch Coefficients in 2 Colour QCD, philosophiæ doctor, National University of Ireland Maynooth, 2015

S. Cotter,Non-perturbative determination of Karsch Coefficients in 2 Colour QCD, philosophiæ doctor, National University of Ireland Maynooth, 2015

work page 2015