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Heavy-ion collision data alone determine the quark-gluon plasma's speed of sound as 0.496c, in agreement with lattice QCD.

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2026-08-02 18:26 UTC pith:RELGCZYW

load-bearing objection A careful, genuinely new extraction of c_s from ATLAS [p_T] data that is transparent about its main assumption — rho=0 at b=0 — but the quoted error does not include that assumption, so the "perfect agreement" with lattice is conditional. the 3 major comments →

arxiv 2603.09647 v2 pith:RELGCZYW submitted 2026-03-10 hep-ph hep-exnucl-exnucl-th

Extracting the speed of sound of QCD from transverse momentum fluctuations

Mubarak Alqahtani , Tribhuban Parida , Jean-Yves Ollitrault This is my paper
classification hep-ph hep-exnucl-exnucl-th PACS 25.75.-q25.75.Nq12.38.Mh
keywords speed of soundquark-gluon plasmatransverse momentum fluctuationsultra-central collisionsheavy-ion collisionshydrodynamic responselattice QCDATLAS
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims to extract the speed of sound of the quark-gluon plasma from the distribution of the transverse momentum per particle in ultra-central lead-lead collisions, using data from the ATLAS experiment. The underlying physics is that the mean transverse momentum rises with collision multiplicity, and the rise rate encodes the square of the speed of sound. The paper corrects two large biases that previously stood in the way: the detector's low-transverse-momentum cutoff and the statistical noise from hadronization. After these corrections, the fit gives cs² = 0.246 ± 0.008, that is cs/c = 0.496 ± 0.008 at an effective temperature of 221 ± 13 MeV, matching first-principles lattice QCD calculations. If the extraction holds, the speed of sound becomes a directly measured property of the quark-gluon plasma rather than a prediction awaiting confirmation.

Core claim

Starting from the effective hydrodynamic relations N ∝ S and [pT] ∝ T_eff with entropy density s(T_eff) ∝ S/(R²)^{3/2}, the paper derives an event-by-event identity δ[pT]/⟨[pT]⟩ = cs² (δS/⟨S⟩ − 3/2 δR²/⟨R²⟩), which ties the variation of mean transverse momentum at fixed multiplicity to the speed of sound. Because ATLAS does not detect particles below 0.5 GeV/c, the paper folds in acceptance factors C_A and D_A constructed from the measured v0(pT) and applies a Bayesian deblurring step to undo the Poisson noise of hadronization. Fitting the mean and relative variance of [pT] as functions of N_ch in ultra-central events, and assuming the plasma size R² is uncorrelated with the entropy S at zer

What carries the argument

The central object is the two-dimensional hydrodynamic response matrix linking the initial-state fluctuations (total entropy S and rms transverse radius R²) to the final-state observables (multiplicity N and mean transverse momentum [pT]). The load-bearing identity is δ[pT]/⟨[pT]⟩ = cs² (δS/⟨S⟩ − 3/2 δR²/⟨R²⟩), which turns the experimentally observed rise of ⟨[pT]⟩ with N into a measurement of cs² once the acceptance factors C_A and D_A (derived from v0(pT)) and the hadronization deblurring are applied. The paper's Appendix A supplies the algebra that propagates the assumption ρ(S,R²)=0 at b=0 through the covariance matrix.

Load-bearing premise

The extraction assumes that the total entropy and the transverse size of the quark-gluon plasma fluctuate independently in collisions at zero impact parameter (ρ=0); if this correlation is actually nonzero, the reported speed of sound shifts by about 1.5 times that correlation, and the paper identifies this as the only irreducible source of uncertainty.

What would settle it

Measure the correlation ρ between total entropy and transverse size at zero impact parameter—for instance, by comparing the charged multiplicity with a size-sensitive observable such as the mean transverse momentum or elliptic flow in events with the same multiplicity. If |ρ| exceeds about 0.03, the quoted cs² would shift by more than its 0.008 error, and the central value as stated would be falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The speed of sound of the quark-gluon plasma near T≈220 MeV is fixed by data at cs/c = 0.496 ± 0.008, in agreement with lattice QCD.
  • The analysis yields a data-driven estimate of initial-state fluctuations: 2.77±0.05% for entropy and 3.09±0.15% for transverse-size fluctuations at zero impact parameter.
  • Because the acceptance corrections are smaller for CMS and ALICE (lower pT cuts), repeating the analysis on their data should produce the same speed of sound, providing a cross-check.
  • The skewness of [pT] fluctuations, already measured, can be modeled along the same lines to learn how the centrality resolution depends on multiplicity.
  • The only irreducible uncertainty is the correlation between S and R² at b=0; constraining that correlation would tighten the result further.

Where Pith is reading between the lines

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

  • The implied initial-state fluctuation amplitudes can serve as a benchmark for collision models; the paper's own simulation requires a gamma-distribution width parameter two times larger than typical Bayesian fits, suggesting that current models may overestimate event-by-event fluctuations.
  • A dedicated measurement of ρ(S,R²) at zero impact parameter, for example by comparing multiplicity with a size-sensitive observable such as elliptic flow, would either confirm the extracted cs or shift it by ≈1.5ρ; this is a concrete near-term test.
  • The same effective-hydrodynamics framework could be applied to other moments (e.g., skewness) to constrain the multiplicity-dependence of centrality resolution, which is currently unknown.
  • If the extraction is repeated at other collision energies or by other experiments and yields the same cs at the same effective temperature, that would confirm the result is a genuine medium property rather than a feature of one collision geometry.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a data-driven extraction of the QCD speed of sound from ATLAS measurements of the mean and variance of the transverse momentum per particle, [p_T], in ultra-central Pb+Pb collisions. The authors construct a linear-response model (Eqs. 2-11) that maps initial-state entropy S and rms radius R^2 onto final-state multiplicity N_A and [p_T^A], with corrections for the low-p_T acceptance cuts using the newly measured v_0(p_T), and a Bayesian unfolding of hadronization (Poisson) fluctuations. Fitting the model to ATLAS data, and assuming that S and R^2 are uncorrelated at zero impact parameter (rho=0), they obtain c_s^2 = 0.246 ± 0.008 (Eq. 25), i.e., c_s/c = 0.496 ± 0.008 at T_eff = 221 ± 13 MeV, in agreement with lattice QCD. The paper is transparent about corrections and error sources, and it explicitly identifies rho=0 as an assumption; however, the quoted uncertainty excludes the effect of a nonzero rho, which the authors themselves estimate shifts c_s^2 by about 1.5 rho.

Significance. If the rho=0 assumption is valid, this is a significant, qualitatively new determination of c_s from ultra-central heavy-ion data, complementary to Bayesian global fits. The paper's treatment of detector acceptance via v_0(p_T) and its systematic deblurring of statistical hadronization fluctuations are valuable methodological contributions. The derivation in Appendix A is clean, and the authors carefully enumerate and quantify most systematic errors. The main caveat is that the central number is conditional on an untested initial-state correlation assumption, and the data analyzed cannot by themselves pin down that assumption; the stated uncertainty is therefore narrower than the true model uncertainty.

major comments (3)
  1. [Sec. VII, Eq. (25)] The quoted result c_s^2 = 0.246 ± 0.008 and the abstract's claim of 'perfect agreement with lattice QCD' are conditional on rho(S,R^2)|_{b=0}=0. The paper itself states that a nonzero rho shifts c_s^2 by about 1.5 rho and calls this the 'only irreducible source of uncertainty,' but this source is not included in Eq. (25) and is not bounded by data. The error bar is therefore a conditional error, not the full uncertainty on the central claim. Please present the result as c_s^2 = 0.246 ± 0.008 (stat+syst under rho=0) ± 1.5 rho, or provide a data-driven bound on rho, and soften the abstract accordingly.
  2. [Appendix A, Eqs. (A4)-(A7)] The inversion from final-state covariances to initial-state covariances is degenerate with respect to c_s^2 and C_12. Equations (A5)-(A6) show that for a given measured set (Sigma_11, |Sigma|, and the conditional mean), different pairs (c_s^2, C_12) can reproduce the same data. Setting C_12(0)=0 is a model input taken from TRENTo/general arguments, not a constraint derived from the ATLAS moments. This is load-bearing because it is the only thing that fixes c_s^2. At minimum, the paper should state explicitly that the reported c_s^2 is not identifiable without this assumption, and the uncertainty should reflect the allowed range of C_12.
  3. [Abstract and Sec. VII] The abstract says the rho=0 scenario is 'preferred both by high-energy QCD and heavy-ion data,' but the text (Sec. VII) supports this with 'general theoretical arguments' [18] and TRENTo model comparisons, while explicitly noting that global theory-to-data comparisons could bound rho but that this is 'beyond the scope' of the work. The empirical support from heavy-ion data is therefore indirect at present. Please rephrase to distinguish model preference from data constraint, and consider adding the suggested rho-sensitivity formula to the abstract.
minor comments (5)
  1. [Sec. VII] Calling rho=0 the 'only irreducible source of uncertainty' is overstated; other model inputs (e.g., the form of the linear response, the acceptance coefficients C_A, D_A, the Gaussianity assumption) are also model-dependent, though they are quantified. 'Unquantified' or 'dominant model uncertainty' would be more precise.
  2. [Eq. (26)] The extracted relative standard deviations of S and R^2 also assume rho=0. If rho is nonzero, the reverse-engineering formulas in Appendix A change; this should be noted alongside Eq. (26).
  3. [Fig. 2] In the caption, 'k2' should be typeset as k^2 for consistency with the text.
  4. [Sec. VIII] The sentence 'It seems likely that the effect of fluctuations would be similar with a hadronic afterburner' is speculative; it would be helpful to label this explicitly as an estimate or to provide a reference.
  5. [References] The footnote in Sec. VI correcting Ref. [3] is useful but slightly buried; consider moving the correction to a footnote at the first use of Eq. (5) or to the acknowledgment.

Circularity Check

0 steps flagged

No circularity: cs² is a fitted free parameter against external ATLAS data, not an input recycled as an output.

full rationale

The central extraction is self-contained. ATLAS provides the mean and variance of [p_T] as functions of N_ch; Eq. (9) is a linear response matrix in which cs² is a free slope-like parameter fitted to these external data (Sec. VII). The acceptance factors C_A, D_A are not fitted to the extracted cs²; they are computed from a hydrodynamic calculation of v0(p_T) (Fig. 1) and benchmarked against ATLAS separately. The hadronization deblurring (Sec. VI) is a mathematical unfolding of Poisson noise with coefficients fixed by N_ch statistics, not by cs². The only significant assumption, ρ(S,R²)=0 at b=0, is an input, not an output: the paper states it explicitly, takes it from TRENTo/general arguments plus global theory-data comparisons, and quantifies the resulting shift as ≈1.5ρ (Sec. VII: 'For cs², there is one irreducible source of uncertainty... the value of the correlation between S and R² at b=0'), so no fitted parameter is renamed as a prediction. Although several supporting references are self-citations (e.g., Refs. [2,3,13,16,18]), the load-bearing relation Eq. (2) is also corroborated by non-overlapping-author simulations (Refs. [4,5]) and the final cs² is compared to, not derived from, lattice QCD. The main caveat is statistical rather than circular: the quoted ±0.008 is conditional on ρ=0, and bounding ρ from data would be needed for the headline uncertainty, but this does not make the derivation equivalent to its inputs.

Axiom & Free-Parameter Ledger

7 free parameters · 7 axioms · 0 invented entities

No new physical entities are introduced. The model rests on hydrodynamic linear-response theory, Gaussian fluctuation assumptions, and Poisson hadronization sampling; the only truly ad hoc ingredient is the untested ρ=0 correlation at b=0. The free parameters are fitted shape functions plus the central cs²; the acceptance factors C_A, D_A and ⟨pT⟩/T_eff are model-derived inputs that partially propagate into the central value.

free parameters (7)
  • cs² (speed of sound squared) = 0.246 ± 0.008
    Fitted to the slope of ⟨[pT_A]|Nch⟩ and the variance k²(Nch), with acceptance corrections; the central output of the paper.
  • A0, A1, A2 for |Σ|/Σ11 parametrization = not quoted individually
    Shape parameters of the b-dependence of the relative variance of [pT_A] at fixed N_A, App. B; fitted to ATLAS mean and variance data and influence the inference of cs.
  • Mean [pT_A] as function of b (rational parametrization coefficients a1, a2, a3) = not quoted
    Free function fitted to ATLAS data, App. B; needed to evaluate the moments before averaging over b.
  • A1/A0 ratio for Σ11 (N_A relative variance) = varied between 1 and 5
    Cannot be constrained from data; assumed to mimic C11. Variation is included in the systematic error.
  • C_A acceptance factor = 0.64 ± 0.01
    Computed in hydrodynamics (Fig. 1), benchmarks against ATLAS data, then used in the linear map. Not fitted to the final cs within this paper, but is a model-derived input.
  • D_A acceptance factor = 0.75 ± 0.05
    Computed in hydrodynamics (Fig. 1); new parameter of the acceptance model.
  • ⟨pT⟩/T_eff ratio = 2.96 (estimated)
    Conversion from cs to a temperature: starts from 3.07 [2], lowered to 3.00 ± 0.05 with afterburner [3], to 3.03 with fluctuations [16], then interpolated down to 2.96 assuming afterburner + fluctuations. This is an estimate, not a full simulation.
axioms (7)
  • domain assumption Hydrodynamic linear response: final fluctuations (δN, δ[pT]) are linearly related to initial fluctuations (δS, δR²) via cs² (Eqs. 4, 9).
    Used throughout; standard effective-hydrodynamics picture, supported by Refs. [2,16,22] and by the paper's own model, but is a modeling assumption, not measured.
  • domain assumption The joint distribution of δS, δR² (and hence δN_A, δ[pT_A]) at fixed b is a centered 2-D Gaussian (Sec. IV).
    Justified as 'Gaussian to a good approximation' by Ref. [10]; central to deriving the conditional moments (Eqs. 10–11).
  • domain assumption The distribution of N_ch at fixed b is Gaussian (Sec. V).
    Needed to reconstruct ⟨N_ch|b⟩ and σ_Nch from P(N_ch) via the Gaussian superposition fit.
  • domain assumption Hadronization noise: for a given N_A, the number of detected tracks obeys a Poisson distribution with mean N_A (Var(N_ch|N_A) = N_A/ε, Eqs. 12–15).
    Standard Monte-Carlo sampling assumption; used to unfold statistical noise and to renormalize the slope by ≈0.81 (Eq. 21).
  • ad hoc to paper R² and S are uncorrelated at b = 0 (ρ(S,R²)|_{b=0} = 0), and their covariance in the b>0 range is taken from TRENTo.
    The central load-bearing assumption, described in Sec. II, VII and App. C: 'Our extraction of the speed of sound relies on the hypothesis that ρ vanishes at b=0.' The paper acknowledges that a nonzero ρ shifts cs² by ~1.5 ρ; it cannot be bounded by data.
  • domain assumption Equation of state: s(T_eff) ∝ S/(R²)^{3/2} and [pT] ∝ T_eff, with proportionality factors identical across events (Eq. 2).
    Dimensional-analysis/hydrodynamic ansatz; its accuracy is discussed in Sec. VIII but is not proved in the paper.
  • domain assumption The shape of the pT spectrum depends only on [pT], enabling Eq. (6) with v0(pT)/v0.
    The 'to a good approximation' decomposition used to derive the acceptance factors (Eqs. 6–8).

pith-pipeline@v1.3.0-alltime-deepseek · 14113 in / 10497 out tokens · 72339 ms · 2026-08-02T18:26:41.313688+00:00 · methodology

0 comments
read the original abstract

We extract the speed of sound ($c_s$) in the quark-gluon plasma from ATLAS data on the probability distribution of the transverse momentum per particle, $[p_T]$, in ultra-central Pb+Pb collisions. With an ideal detector, $c_s$ can be inferred from the rise of the mean $[p_T]$ with the collision multiplicity. In practice, however, low-$p_T$ particles escape detection, which biases the analysis. We show how to correct for this bias by using data on the variance of $[p_T]$, as well as information from the recently-measured $v_0(p_T)$. We also introduce a systematic method for deblurring the noise from the hadronization process. Assuming that the size of the quark-gluon plasma is independent of the hadron multiplicity in collisions at zero impact parameter, which is the scenario preferred both by high-energy QCD and heavy-ion data, we obtain $c_s/c=0.496\pm 0.008$ at temperature $T=221\pm 13$~MeV, in perfect agreement with first-principles calculations from lattice QCD.

Figures

Figures reproduced from arXiv: 2603.09647 by Jean-Yves Ollitrault, Mubarak Alqahtani, Tribhuban Parida.

Figure 1
Figure 1. Figure 1: FIG. 1. Variation of the acceptance factors [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Symbols are ATLAS data [9], lines are fits. (a) Prob [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison between the value of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Line: Variation of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

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Forward citations

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