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REVIEW 3 major objections 5 minor 5 cited by

Thermodynamics of hot strong-interaction matter from ultrarelativistic nuclear collisions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single momentum observable fixes the temperature of the quark-gluon plasma created in heavy-ion collisions.

desk verdict A genuinely new method for extracting the QGP effective temperature and speed of sound from LHC data, but the entropy-density 'confirmation' of lattice QCD is largely circular. read the letter →

arxiv 1908.09728 v2 pith:GCPUYM7I submitted 2019-08-26 nucl-th hep-phnucl-ex

Fernando G. Gardim , Giuliano Giacalone , Matthew Luzum , Jean-Yves Ollitrault This is my paper · ORCID
classification nucl-thhep-phnucl-ex
keywords quark-gluonplasmaheavy-ioncollisionseffectivetemperaturelatticeQCDequationofstatespeedsoundentropydensitymeantransversemomentum
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 claims the first determination of a temperature and a second thermodynamic quantity from heavy-ion collision data, using the measured mean transverse momentum to read off an effective temperature. It derives $T_{\rm eff} = 222 \pm 9$ MeV in central lead-lead collisions at 5.02 TeV, and from the same data an entropy density $s(T_{\rm eff}) = 20 \pm 5$ fm$^{-3}$ and a squared speed of sound $c_s^2 = 0.24 \pm 0.04$. These values agree with lattice QCD calculations, which is why a careful reader would care: a direct, almost parameter-free experimental thermometer for the deconfined phase would establish that the matter created in these collisions is a quark-gluon plasma with the expected number of degrees of freedom.

What carries the argument

The central object is the effective temperature $T_{\rm eff}$ and effective volume $V_{\rm eff}$, defined by $E = \epsilon(T_{\rm eff}) V_{\rm eff}$ and $S = s(T_{\rm eff}) V_{\rm eff}$ from the integrals of the stress-energy tensor and entropy current over the freeze-out hypersurface. Taking the ratio $E/S$ eliminates $V_{\rm eff}$, so $T_{\rm eff}$ is fixed by the equation of state alone. The load-bearing identity is the hydrodynamic proportionality $\langle p_t\rangle = 3.07\, T_{\rm eff}$ with the soft lattice equation of state, which converts a measured momentum into a temperature, and the thermodynamic identity $c_s^2 = d\ln\langle p_t\rangle/d\ln(dN_{\rm ch}/d\eta)$ for the speed of sound.

What would settle it

Compare the speed-of-sound value obtained from the ratio of percent changes in mean transverse momentum and multiplicity across a scan of collision energies to the lattice QCD prediction: a mismatch larger than the stated 0.04 uncertainty at any one energy would invalidate the universal calibration.

Watch

Extended reading notes

Core claim

The paper establishes that the average transverse momentum of charged particles, $\langle p_t\rangle$, is proportional to an effective temperature $T_{\rm eff}$ defined by equating the total energy and entropy at freeze-out to those of a uniform fluid at rest, using the same lattice-QCD equation of state as in the hydrodynamic simulation. In simulations with a soft equation of state, the proportionality constant is $\langle p_t\rangle/T_{\rm eff} = 3.07$ and is nearly insensitive to shear or bulk viscosity, centrality, and collision energy. Using the measured $\langle p_t\rangle = 681$ MeV therefore yields $T_{\rm eff} = 222 \pm 9$ MeV, and combining the charged multiplicity with the effective volume gives $s(T_{\rm eff}) = 20 \pm 5$ fm$^{-3}$; forming the logarithmic derivative of $\langle p_t\rangle$ with respect to multiplicity between two collision energies gives $c_s^2 = 0.24 \pm 0.04$. The paper presents these results as agreeing with ab initio lattice QCD and as evidence that a deconfined phase with active colour degrees of freedom is produced.

Load-bearing premise

The measured average sideways momentum of the produced particles is assumed to be locked to the true temperature by a factor of 3.07, a lock calibrated in one computer model of the collision; if that model's equation of state or freeze-out description is wrong for the real plasma, every extracted number shifts.

Editorial extensions

If this is right

  • The measured temperature $T_{\rm eff}=222\pm9$ MeV ($2.58\times10^{12}$ K) gives an experimental anchor for the quark-gluon plasma equation of state.
  • The entropy density $s/T^3 = 14\pm3.5$, compared with lattice QCD, indicates roughly 30 active degrees of freedom, i.e., deconfined quarks and gluons.
  • The speed of sound $c_s^2=0.24\pm0.04$, about half the speed of light squared, constrains the equation of state in the crossover region.
  • Because $T_{\rm eff}$ is nearly independent of centrality, all centralities at 5.02 TeV probe essentially the same temperature, simplifying comparisons across collision systems.
  • If $\langle p_t\rangle$ were measured at lower beam energies, the same derivative method would extend the thermodynamic curve to lower temperatures and nonzero baryon chemical potential.

Reading between the lines

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

  • If the 3.07 calibration is universal, this turns a single high-statistics observable into a thermometer, and the same procedure could be applied to smaller systems such as proton-nucleus collisions where hydrodynamic analyses are less settled.
  • The energy-derivative route to $c_s^2$ suggests a direct experimental search for a softest point in the equation of state: scan $\langle p_t\rangle$ as a function of collision energy and look for a minimum in its slope with respect to multiplicity.
  • A test worth checking is whether the constant 3.07 survives a different hydrodynamic code or a different tuned initial-condition model; if it shifts, the quoted temperature would move with it, even though the paper's internal checks cover shear viscosity, bulk viscosity, and the stiff equation of state.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a method to extract the effective temperature, entropy density, and speed of sound of the quark-gluon plasma produced in ultrarelativistic heavy-ion collisions. Using hydrodynamic simulations with the soft lattice-QCD equation of state s95p-v1, the authors establish that the mean transverse momentum of charged hadrons is proportional to an effective freeze-out temperature, <pt> = 3.07 Teff, across centralities, collision energies, and different transport coefficients. They then combine this calibration with the measured <pt> = 681 MeV to obtain Teff = 222 ± 9 MeV, use the charged-particle multiplicity and the hydrodynamic effective volume to obtain s(Teff) = 20 ± 5 fm^-3 (i.e., s/T^3 = 14 ± 3.5), and use the relative variation of <pt> and dNch/dη between 2.76 and 5.02 TeV to obtain c_s^2 = 0.24 ± 0.04. The paper concludes that these results agree with lattice QCD and confirm the production of a deconfined phase.

Significance. If the extraction is robust, the paper offers an unusually direct experimental route to the thermodynamics of the quark-gluon plasma, with the attractive feature that many model-dependent normalization factors cancel in ratios. The systematic exploration of ideal, shear-viscous, and bulk-viscous hydrodynamics, as well as the use of both soft and stiff equations of state, is a real strength, and the authors make their hydrodynamic code publicly available. The speed-of-sound extraction from the energy dependence of <pt> and multiplicity is elegant and is the most model-independent part of the paper. However, the entropy-density result is largely circular as presented, and the quoted uncertainties do not yet include the equation-of-state sensitivity of the central calibration. These issues affect the main quantitative claims and therefore need to be addressed before the paper can be accepted.

major comments (3)
  1. [Eq. (1) and Eq. (4), Fig. 4 (top)] The reported s(Teff)/T^3 = 14 ± 3.5 is not an independent confirmation of lattice QCD as it stands. In the simulation, Veff is defined through S = s_EOS(Teff) Veff using the s95p-v1 equation of state, and the initial entropy is normalized centrality-by-centrality to the measured dNch/dη. When Eq. (4) is evaluated with this hydrodynamic Veff and the measured multiplicity, it approximately returns the input s95p-v1 entropy density, provided the hydrodynamic entropy per charged particle agrees with the assumed S/Nch = 6.7. The manuscript should report this consistency check explicitly (e.g., compare the hydrodynamic S/Nch with 6.7) and state which part of the extracted s is genuinely independent of the input equation of state. As written, the top panel of Fig. 4 demonstrates consistency of the simulation with its own equation of state rather than an ab-initio confirmation.
  2. [Eq. (3) and Fig. 2 (dot-dashed lines)] The quoted uncertainty Teff = 222 ± 9 MeV includes the 4% freeze-out temperature variation but excludes the 6% change in <pt>/Teff obtained with the stiff equation of state. The argument that the stiff equation of state overpredicts <pt> by 40% disfavors that particular equation of state, but it does not bound the calibration ratio for other equations of state that still reproduce the measured <pt>. Because Teff enters all subsequent thermodynamic extractions, the paper should either include an equation-of-state modeling contribution in Eq. (3) or provide a calibration test over a range of equations of state that remain compatible with the data. Without this, the uncertainty on Teff is incomplete.
  3. [Eqs. (6)-(8)] The speed-of-sound extraction c_s^2 = 0.24 ± 0.04 assumes that the proportionality coefficients in Eq. (6) are energy-independent between 2.76 and 5.02 TeV. The authors estimate the Veff variation (at most 3%) but do not propagate the equation-of-state sensitivity of the <pt>-to-Teff normalization across this energy interval. Since the stiff-equation-of-state test changes this normalization by 6% in absolute value, a relative change of similar magnitude between the two energies could shift the derived c_s^2 well beyond the quoted ±0.04. Please show the stiff-equation-of-state predictions for <pt>/Teff at both 2.76 and 5.02 TeV, or otherwise justify that this effect is negligible.
minor comments (5)
  1. [Fig. 2 caption] The black and red curves are plotted on different vertical scales that differ by the factor 3.07, so the statement in the text that 'black lines and red lines overlap' is only true after rescaling; please clarify this in the caption to avoid confusion.
  2. [Methods] The approximation dN/dy ≃ 1.15 dN/dη is used without derivation or an error estimate; a short justification or reference would improve the reproducibility of Eq. (5).
  3. [Fig. 4 bottom] The grey box is shifted 5 MeV lower than the Teff of Eq. (3) assuming a linear interpolation of the temperature between 2.76 and 5.02 TeV; please state this assumption explicitly.
  4. [Abstract and introduction] The phrase 'the first such determination' should be tempered or carefully justified, since thermal fits (Ref. [24]) already provide freeze-out temperature estimates; the novelty is better described as a simultaneous determination of the temperature with a second thermodynamic quantity from the same data.
  5. [Code Availability] The code availability statement links to the generic MUSIC repository but not to the specific configuration files, initial conditions, or scripts used for this paper; providing these would make the results fully reproducible.

Circularity Check

1 steps flagged · score 6.0 of 10

The s/T^3 "agreement with lattice QCD" is largely an echo of the lattice-based EOS used to define Veff; the speed-of-sound result remains independent.

  1. self definitional [Eq. (1), Eq. (4), and Fig. 4 (top)]
    "defined by the equations E = ∫ f.o. T^{0μ} dσμ = ε(Teff)Veff, S = ∫ f.o. s u^{μ} dσμ = s(Teff)Veff ... By taking the ratio E/S, one eliminates Veff, and one can solve the resulting equation for Teff, using the same equation of state as in the hydrodynamic calculation. Note that Teff and s(Teff) are related by the equation of state of the fluid by construction."

    In the simulation, Eq. (1) fixes Veff = S_hydro / s_EOS(Teff_hydro). Because the initial entropy is normalized to the measured charged multiplicity and S is conserved in ideal hydro, S_hydro ≈ (S/Nch)_hydro dNch/dy. Substituting this Veff into Eq. (4) gives s(Teff) = [(S/Nch)_data/(S/Nch)_hydro] s_EOS(Teff_hydro), i.e. the "measured" entropy density is the s95p-v1 lattice-based EOS (the one put into MUSIC) rescaled by the ratio of entropy-per-charged-particle estimates. With S/Nch = 6.7 from Ref. [25] close to the simulation value, the reported s/T^3 ≈ 14 and the "agreement with ab-initio calculations" in Fig. 4 (top) mostly return the input EOS rather than independently confirming lattice QCD. The c_s^2 result from Eq.

full rationale

The central Teff extraction is model-calibrated rather than circular: experimental <pt> is combined with a hydrodynamically computed ratio <pt>/Teff = 3.07, and the temperature itself is not an input of the simulation. However, the claimed independent confirmation of lattice QCD in the entropy-density panel is partly circular. By Eq. (1), Veff is defined through the same equation of state used in the hydrodynamic run, and Eq. (4) divides the data-determined entropy by this Veff; if the simulation's entropy per charged particle matches the external value 6.7, s(Teff) reduces to the s95p-v1 EOS evaluated at Teff. Since s95p-v1 is itself a lattice-QCD-based parameterization, the top panel of Fig. 4 compares input with input. The speed-of-sound determination from d ln<pt>/d ln(dNch/dη) is genuinely more independent, as the paper notes that Veff and the proportionality coefficient cancel; it is however validated only under the soft-EOS dynamics and does not rescue the s/T^3 claim. Author self-citations (Refs. [21]-[23]) are used for initial-condition tuning and supporting scalings but are not load-bearing for the main reduction identified above.

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

The central extraction rests on several standard hydrodynamic modeling ingredients: an initial condition model, a fixed freeze-out temperature, transport coefficients from prior Bayesian fits, and the lattice QCD equation of state. The only quantity genuinely fitted to data in this paper is the per-centrality entropy normalization. The universality of <pt>/Teff is a simulation result, not a fit, but it is a model-dependent assumption when applied to nature. No new particles or forces are introduced.

free parameters (5)
  • Initial entropy normalization per centrality = Adjusted to match dNch/deta
    In Methods, the proportionality factor between s(tau0,r) and sqrt(TA TB) is tuned at each centrality to reproduce the ALICE multiplicity. This fit affects Veff and indirectly the entropy density result.
  • Freeze-out temperature Tf.o. = 156.5 MeV default; 150 to 160 MeV variation
    Chosen from hadronic chemical freeze-out fits. The paper varies it to estimate the 4% uncertainty on <pt>/Teff, which sets the Teff error bar.
  • Shear viscosity over entropy ratio eta/s = 0.2 (constant)
    Input from Bayesian analyses of collective flow. One of three hydrodynamic variants used to demonstrate robustness of the <pt>/Teff ratio.
  • Bulk viscosity parametrization = As in Bernhard et al. (2016)
    Used in the bulk-viscous variant. It is a model input from a Bayesian fit, not derived here.
  • Fluctuation correction to initial radius = 5%, 15%, 30% at b = 2, 7, 12 fm
    Hand-tuned shrinkage of smooth initial conditions to match the radius of the TRENTo model with fluctuations. This ad hoc correction changes Veff and hence the entropy density.
assumptions (6)
  • domain assumption Boost-invariant longitudinal expansion
    Invoked in Methods. Standard at LHC midrapidity, but an approximation that neglects finite longitudinal dynamics.
  • domain assumption Negligible baryon chemical potential
    Stated in Methods. Good at LHC energies, but not exact.
  • domain assumption Hydrodynamics is valid from tau0 = 0.6 fm/c
    Thermalization is assumed to be complete by this time; the pre-equilibrium phase is not modeled, only a correction to the system size.
  • domain assumption Initial entropy density proportional to sqrt(TA TB)
    Motivated by the TRENTo model, not derived from QCD. This choice affects the initial radius and Veff.
  • domain assumption Cooper-Frye freeze-out with quadratic viscous corrections and no hadronic rescattering
    The standard freeze-out prescription used in the simulations. The neglect of the hadronic phase is a known simplification.
  • domain assumption Universality of the <pt>/Teff ratio across real collision systems
    The paper demonstrates this universality within its simulations, but assumes it extends to the actual quark-gluon plasma. This is the key modeling leap behind the temperature extraction.

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Cite this review

Pith. "Pith review of Thermodynamics of hot strong-interaction matter from ultrarelativistic nuclear collisions." pith.science (2026). https://pith.science/paper/GCPUYM7I

@misc{pith2026190809728,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of hot strong-interaction matter from ultrarelativistic nuclear collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GCPUYM7I}},
  note         = {Machine review of arXiv:1908.09728}
}
read the original abstract

Collisions between heavy atomic nuclei at ultra-relativistic energies are carried out at particle colliders to produce the quark-gluon plasma, a state of matter where quarks and gluons are not confined into hadrons, and colour degrees of freedom are liberated. This state is thought to be produced as a transient phenomenon before it fragments into thousands of particles that reach the particle detectors. Despite two decades of investigations, one of the big open questions is to obtain an experimental determination of the temperature reached in a heavy-ion collision, and a simultaneous determination of another thermodynamic quantity, such as the entropy density, that would give access to the number of degrees of freedom. Here we obtain the first such determination, utilizing state-of-the-art hydrodynamic simulations. We define an effective temperature, averaged over the space-time evolution of the medium. Then, using experimental data, we determine this temperature, the corresponding entropy density and speed of sound in the matter created in lead-lead collisions at the Large Hadron Collider. Our results agree with first-principles calculations from lattice quantum chromodynamics and confirm that a deconfined phase of matter is indeed produced.

Figures

Figures reproduced from arXiv: 1908.09728 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic representation of a [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Results from hydrodynamic simulations of Pb+Pb collisions. Black curves correspond to the average transverse [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Estimate of the theoretical uncertainty on the effec [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Thermodynamic properties of hot strong-interaction [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

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