{"id":"446e35dc-6184-4c26-b3d8-e35f8e5722b5","arxiv_id":"2412.01467","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The square speed of sound in high-energy collisions is inferred from the Gaussian rapidity width via the Landau model, yielding values mostly from 0 to 1/3.","lead":"This paper uses the Landau hydrodynamic model to connect the width of hadron rapidity distributions to the square speed of sound in high-energy collisions. It argues that in most cases the extracted value falls between 0 and 1/3, with larger values explained by early-stage geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central 0-to-1/3 range rests on the asserted identification sigma = L/6 (Eq. 9), which is neither derived nor compared with measured rapidity widths; changing this coefficient shifts c_s^2 enough to break the claim.","rationale":"The reader identified Eq. (9), sigma = L/6, as the weakest assumption, and I agree that this is the single most load-bearing step. The paper's quantitative conclusion that c_s^2 lies in [0, 1/3] in most cases follows algebraically from this identification together with the scanned k values, but the identification itself is not derived from Landau hydrodynamics, not fitted to data, and not independently substantiated. The text even acknowledges that the relationship is 'proposed' rather than proven, and that larger sigma scenarios 'lack physical justification.' The paper also states that no data sets were generated, and the figures show deterministic curves without experimental points. Therefore the label 'extracted' is an overclaim: the paper is a parametric illustration, not an extraction. My concern is not that the algebra is wrong—Eqs. (10)-(12) are consistent—but that the central claim depends on an arbitrary coefficient. A concrete test using measured rapidity widths would settle whether L/sigma = 6 holds in reality, and a simple sensitivity check shows the claim's fragility. For these reasons I concur with the reader's REJECT verdict: the central claim is not established as stated, though the paper could be reframed as a conditional model study.","tokens_in":9668,"tokens_out":3419,"duration_ms":30850,"concrete_test":"Test sigma = L/6 against data: analyze measured charged-particle rapidity or pseudorapidity distributions in pp and Pb-Pb collisions at sqrt(s_NN) = 0.9, 2.76, 5.02, and 13 TeV, fit a single Gaussian to obtain sigma, and compute L from Eqs. (1)-(8) using leading-nucleon energy-loss measurements (or the same k scan). If L/sigma deviates from 6 by more than ~20% at any energy, Eq. (9) is not generally valid. As a simpler computational check, recompute c_s^2 at sqrt(s_NN) = 200 GeV with sigma = L/6 and sigma = L/4 for k = 0.5-0.9; if the value crosses 1/3, the headline range is coefficient-dominated rather than a physical extraction.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's headline claim, that the extracted square speed of sound lies between 0 and 1/3 in most cases, is controlled by the unvalidated proportionality sigma = L/6 in Eq. (9). This equation equates the kinematic rapidity separation of leading nucleons, L, with the 6-sigma span of a Gaussian rapidity distribution of produced hadrons. That is a statistical convention applied as a physical identification: it assumes the leading nucleons sit at exactly ±3 sigma of the fireball's Gaussian rapidity plateau. The text itself concedes the fragility, stating that Figure 3 scenarios with sigma > L/6 'lack physical justification' and that Eq. (9) fails if the rapidity distribution is multi-Gaussian. No experimental rapidity distribution is fitted in this paper, so sigma is never actually extracted; it is set by Eq. (9) from L, which in turn depends on the scanned energy-loss parameter k. The paper also cites its own prior works [39, 40] as the basis for L = 6 sigma, making that step self-referential rather than independently derived. Because c_s^2 scales roughly as sigma^2 at small sound speed (Eq. 10), a change from sigma = L/6 to sigma = L/4 changes c_s^2 by a factor of about 2.25, enough to push many cases above 1/3. Thus the central claim is not robust; it is an artifact of an assumed proportionality constant.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reviews the square speed of sound c_s^2 across different forms of matter and then proposes a method to extract c_s^2 for hadronic matter decoupled from the hot, dense system in high-energy collisions. Assuming an energy-loss rate k for incident nucleons (Eqs. 1-4), the authors compute the total rapidity shift L between the leading target and projectile nucleons (Eq. 8). They then equate L with the 6-sigma span of a Gaussian rapidity distribution of produced hadrons (Eq. 9), set sigma = L/6, insert this into the Landau-model relation between sigma and c_s^2 (Eq. 10), and solve for c_s^2 (Eqs. 11-12). Scanning k over 0.5-0.9 and sqrt(s_NN) over a wide range, they find that c_s^2 lies between 0 and 1/3 in most cases (Fig. 2); larger sigma values (Fig. 3) produce c_s^2 above 1/3, which the authors disclaim as lacking physical justification. The paper closes with qualitative explanations for c_s^2 = 1/d based on the dimensionality of expansion and dilution of the produced matter.","tokens_in":9920,"tokens_out":11551,"duration_ms":92815,"significance":"If the central claim were established, the paper would offer a simple kinematical route to c_s^2 from the measured width of rapidity distributions, connecting the conformal bound c_s^2 <= 1/3 to a Gaussian-width convention. The algebraic inversion of Eq. (10) in Eqs. (11)-(12) is correct, and the manuscript is commendably candid about its failure modes: it explicitly states that the Figure 3 scenarios lack physical justification and that Eq. (9) cannot be used when the rapidity distribution is a superposition of multiple Gaussians. The literature survey across neutron stars, dark matter, and dark energy is informative. However, the headline claim is not an extraction: no experimental rapidity distribution is fitted, and the 0-to-1/3 range is a direct consequence of the asserted normalization sigma = L/6 rather than of measured data.","major_comments":[{"comment":"The identification sigma ≈ L/6 is asserted on the basis of the Gaussian 99.7% rule, but this is a statistical convention rather than a physical relation. Equating the kinematic rapidity separation of the leading nucleons with the 6-sigma span of the produced-hadron distribution is neither derived nor tested against any measured rapidity distribution in this paper. Because Eq. (10) gives c_s^2 proportional to sigma^2 at small c_s^2, changing the coefficient from 1/6 to 1/4 changes c_s^2 by a factor of about 2.25; the paper's own Fig. 3 shows that such a change pushes most values above 1/3. The range 0 to 1/3 in Fig. 2 is therefore a property of the chosen normalization, not an empirical result, and the abstract's phrase 'The extracted square speed of sound lies within a range from 0 to 1/3' overstates the support the analysis provides.","section":"Sec. 2, Eq. (9)"},{"comment":"The energy-loss parameter k is scanned over 0.5-0.9 on the basis of proton-nucleus studies [23-26], but no comparison is made with measured leading-nucleon spectra or with measured rapidity widths at the energies considered, and the Data Availability Statement confirms that no data sets were used. Since L and hence sigma are determined by k through Eqs. (1)-(9), both the magnitude of c_s^2 and the energy at which c_s^2 crosses 1/3 depend on the assumed k interval. The statement that c_s^2 lies in 0 to 1/3 'in most cases' is consequently a statement about the model's parameter space rather than about collisions, unless the k range is anchored to data.","section":"Sec. 3, Figs. 1-2 and Data Availability Statement"},{"comment":"The manuscript states that the connection between L and c_s^2 is established via L = 6 sigma using the authors' previous works [39,40]. Those works fitted (pseudo)rapidity distributions and already assumed the Landau relation Eq. (10); the fits are not reproduced here. The only genuinely new step in the present paper is Eq. (9), and that is precisely the step that fixes the claimed 0-to-1/3 range. The central claim is therefore not independently validated in this manuscript, and the reliance on the self-cited earlier analyses makes the argument appear self-referential.","section":"Sec. 3, discussion of refs. [39,40]"},{"comment":"The text concedes that the sigma > L/6 scenarios 'lack physical justification' and that, once the rapidity distribution is a superposition of two or three Gaussians, 'each sigma can no longer be obtained from the relationship with L' via Eq. (9). These concessions directly limit the claimed universality of the 0-to-1/3 range, yet the Summary restates the range as a robust extracted result with only qualitative caveats. The manuscript should either restrict its claims to the conditional statement 'under the assumptions of Eq. (9),' or supply direct fits to measured rapidity distributions that support the range as an extraction.","section":"Sec. 3, paragraphs after Fig. 3 and Sec. 4 Summary"}],"minor_comments":[{"comment":"The notation uses awkward double-letter subscripts (E_TT_beam, E_LT, etc.) throughout, apparently from the original formatting; this harms readability and should be replaced with standard single-letter subscripts. The beam frame (laboratory vs. center-of-mass) should also be defined explicitly before Eq. (5).","section":"Sec. 2, Eqs. (1)-(5)"},{"comment":"The binary-collision number nu is introduced with quoted fractions from [23] to motivate the k range, but nu is never used quantitatively; the connection between the nu distributions and the chosen values k = 0.5-0.9 should be made explicit.","section":"Sec. 3, paragraph before Fig. 1"},{"comment":"The claim that the non-conformal viscous solution of [22] yields 'a marginally larger sigma^2' compared with the conformal solution is stated without the relevant formula or quantitative comparison; please provide the relationship or the numerical values.","section":"Sec. 2, discussion of ref. [22]"},{"comment":"Eq. (10) is presented as the Landau-model relation without derivation; a brief derivation or a precise pointer to the expression in refs. [16,21] would clarify the origin of the factor (8/3) c_s^2 / (1 - c_s^4) and the model's domain of validity.","section":"Sec. 2, Eq. (10)"},{"comment":"The final paragraph interprets values between 1/3 and 1/2 as indicating non-central collisions and values between 1/2 and 1 as indicating central collisions, but no quantitative argument is given for these thresholds; given that Fig. 3 values above 1/3 are disclaimed as physically unjustified, this explanatory scheme appears to be in tension with the rest of the paper.","section":"Sec. 4, Summary"},{"comment":"There are many typographical and grammatical errors (e.g., 'leadings to', 'slowly decelerate', inconsistent use of 'square speed of sound' instead of the more standard 'squared speed of sound'); a thorough copy-edit is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The manuscript reads as a mini-review with a parametric study rather than a data-driven analysis. The headline claim of a 0-to-1/3 range for the extracted c_s^2 derives from the unvalidated identification sigma = L/6 in Eq. (9), and the paper's own text concedes the fragility of that step. The authors' prior works [39,40], which did fit data, are used to legitimate the central step without reproducing the fits, so the novelty claim rests on a self-referential basis; this citation pattern is worth the editor's attention. Even with a substantial reframing to a 'model-conditional statement,' the remaining content would be a Gaussian-convention argument plus a literature survey, which is thin for a high-energy physics journal. I recommend rejection, though the paper might be suitable for a venue that explicitly publishes pedagogical or parametric notes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Honestly, this paper is a well-written parametric exercise that overlabels itself. The core relationship between sigma and c_s^2 is textbook Landau; the sigma = L/6 link is carried over from the authors' own earlier papers, and the only real new work here is a scan over the energy-loss parameter k and over alternative sigma/L values. The paper also includes a useful, compact survey of c_s^2 ranges in other contexts (neutron stars, dark energy, etc.), which is probably its most solid part.\n\nWhat it does well: the algebra in Eqs. (10)-(12) is correct, and the authors are honest about the fragility of the central assumption. They explicitly write that the sigma values in Figure 3 'lack physical justification,' and they acknowledge that Eq. (9) fails if the rapidity distribution is multi-Gaussian. That candor takes some of the sting out of the self-citation.\n\nThe soft spots are structural. The claim that 'the extracted square speed of sound lies within a range from 0 to 1/3 in most cases' is not an extraction. No measured rapidity distribution is fitted anywhere; sigma is not obtained from data. Instead, sigma is set to L/6, and L depends on the assumed k. The 0-to-1/3 range is a direct consequence of that choice. The paper even demonstrates the sensitivity: with sigma = 0.2L (only slightly larger than L/6), most curves exceed 1/3. So the headline range is closer to an artifact of the normalization than a discovery about the conformal bound.\n\nThe dimensional arguments for c_s^2 > 1/3 are speculative and untested. They might be interesting as a heuristic, but they are not supported by any evidence in this paper.\n\nWho is this for? Someone who wants a quick overview of c_s^2 ranges across different physical settings and a simple illustration of how Landau hydrodynamics plus a Gaussian width assumption can produce a range. As a research contribution, it doesn't hold up, and I wouldn't cite it.\n\nMy honest recommendation: if this came to me as an editor, I would desk reject it on the grounds that the central claim is not supported by the evidence presented; but if you want to be generous, you could send it to a secondary journal as a pedagogical note, with the explicit request that the authors reframe it as a parametric study and drop the word 'extracted.' Not a serious referee target for a main journal.","headline":"A well-organized parametric study that overclaims: the 0-to-1/3 range is a consequence of the assumed sigma = L/6, not an extraction from data.","tokens_in":10536,"tokens_out":3047,"would_cite":false,"duration_ms":24990,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.40.Ee","13.85.Ni","13.87.Ce"],"model":"deepseek-v4-flash","headline":"This paper argues that the square speed of sound in hadronic matter decoupled from high-energy collisions, extracted from Gaussian rapidity distributions via the Landau relation, lies between 0 and 1/3 for most collision energies.","keywords":["square speed of sound","energy loss","rapidity shift","Gaussian rapidity distribution","Landau hydrodynamic model","hadronic matter","conformal bound","high-energy collisions"],"falsifier":"Measure the full rapidity distribution in proton-proton collisions at, say, 200 GeV and 13 TeV, fit it to a single Gaussian, and compute $L$ from leading-nucleon energies using Eqs. (1)-(8). If the fitted $\\sigma/L$ differs from $1/6$ by more than experimental uncertainty, or if the $c_s^2$ extracted from Eq. (12) disagrees with the value obtained from lattice-QCD equation-of-state calculations under the same conditions, the central claim fails.","tokens_in":9385,"feed_emoji":"🔊","tokens_out":12948,"duration_ms":100401,"temperature":0.7,"pith_summary":"The paper offers a simple way to extract the square speed of sound, $c_s^2$, of the hadronic matter that freezes out in high-energy collisions. It treats the produced particles' rapidity distribution as one Gaussian, computes the rapidity gap $L$ between leading target and projectile nucleons after an assumed energy-loss fraction $k$ between 0.5 and 0.9, and sets the Gaussian width to $\\sigma = L/6$, the six-$\\sigma$ range of a Gaussian. Plugging this width into the Landau hydrodynamic relation converts it into $c_s^2$, which the paper finds stays between 0 and 1/3 for most nucleon-nucleon center-of-mass energies, with values above 1/3 appearing only near $\\sqrt{s_{NN}}\\sim 10$ TeV. Values above 1/3 are interpreted as early, not-yet-expanded cylindrical geometries that later cool down below 1/3. If the extraction works, a rapidity-width measurement plus an energy-loss assumption reproduces the conformal bound without lattice or equation-of-state input.","feed_headline":"Sound speed from hadron data mostly obeys 0 to 1/3","feed_subtitle":"A rapidity-width analysis plus Landau hydrodynamics reproduces the conformal bound for most collision energies.","key_machinery":"The load-bearing object is the Landau relation between the Gaussian width $\\sigma$ of the rapidity spectrum and the square speed of sound, $\\sigma^2 = \\frac{8}{3}\\frac{c_s^2}{1-c_s^4}\\ln\\left(\\frac{\\sqrt{s_{NN}}}{2m_N}\\right)$, together with the inversion that yields $c_s^2$ from $\\sigma$ and $\\sqrt{s_{NN}}$. To use this relation, the paper sets $\\sigma \\approx L/6$ via the 99.7% containment of a Gaussian, where $L$ is the rapidity shift between leading target and projectile nucleons computed from Eqs. (1)-(8) with an assumed energy-loss fraction $k$ between 0.5 and 0.9. The six-$\\sigma$ identification is what turns a kinematic statement about leading nucleons into a width for produced hadrons, and the assumed $k$ window sets how large $L$ is at each energy.","core_discovery":"The paper's central claim is that the square speed of sound, $c_s^2$, of hadronic matter that has decoupled in high-energy collisions can be read off from the width of a single Gaussian rapidity distribution through the Landau relation, and that when the width is fixed to one sixth of the total rapidity shift between leading target and projectile nucleons, the extracted values fall in the range $0 \\le c_s^2 \\le 1/3$ for most collision energies. The extraction chain is explicit: beam energies and an assumed energy-loss fraction $k$ in the interval 0.5 to 0.9 give the leading-nucleon rapidities, their difference gives $L$, the 99.7% containment rule of a Gaussian sets $\\sigma = L/6$, and the Landau relation converts $\\sigma$ into $c_s^2$. Below roughly ten TeV the curves stay under $1/3$; around and above that scale some cases cross $1/3$. The paper attributes values from $1/3$ to $1/2$ to non-central collision cylinders with transverse flow, values from $1/2$ to $1$ to central cylinders, and states that all such values fall below $1/3$ once the system expands. It also states that the larger-width curves in Figure 3 are not physically realizable, and that at very high energies a single Gaussian may give way to two or three Gaussian sources, each with its own $c_s^2$.","pith_inferences":["Not addressed by the paper: the 0-to-1/3 window is tied to the six-sigma convention; choosing a different containment range would shift all extracted values and change the crossing energy.","Not addressed by the paper: direct measurement of leading-baryon energy loss would replace the assumed $k$ window with data, and an energy-dependent $k$ would move the predicted $\\sim 10$ TeV crossing point.","Not addressed by the paper: applying the same width-based extraction separately to pions, kaons, and protons could reveal species-dependent $c_s^2$, since each species has its own rapidity width."],"forward_implications":["If the extraction holds, a single Gaussian fit to rapidity spectra plus a 50-90% energy-loss assumption reproduces $c_s^2 \\le 1/3$ for most collision energies without any equation-of-state input.","The predicted crossing of $1/3$ near $\\sqrt{s_{NN}}\\sim 10$ TeV gives a specific energy where forward-rapidity data can test the scheme.","At very high energies the single Gaussian is expected to fail, so two- or three-source decompositions replace it; the central source has the largest $c_s^2$, and no fourth source is needed at currently accessible energies.","Values above $1/3$ are not violations but early-time cylindrical geometries, $c_s^2=1/d$ with $d=2$ or $d=1$, which later expansion reduces below $1/3$."],"supporting_citations":[{"why":"Establishes the Landau hydrodynamic picture in which rapidity spectra are Gaussian and the total rapidity shift defines the distribution range.","marker":"[11–14]"},{"why":"Provides the Landau relation between the Gaussian width and the square speed of sound used in Eq. (10).","marker":"[11, 16–21]"},{"why":"Gives a non-conformal viscous Landau solution with an alternative sigma-squared/sound-speed relation and confirms slightly different rapidity widths.","marker":"[22]"},{"why":"Source of the 50% single-collision and 10-25% multi-collision energy-loss values that justify the assumed $k$ window 0.5-0.9.","marker":"[23–26]"},{"why":"Supports the dimension argument $c_s^2 = 1/d$ used to interpret values above 1/3.","marker":"[27, 28]"},{"why":"Previous application of the Landau-based extraction to Au-Au and Cu-Cu pseudorapidity distributions, giving $c_s^2$ near 1/3-1/2 for central and non-central regions.","marker":"[39]"},{"why":"Earlier extraction of $\\sigma$ and $c_s^2$ in proton-proton and proton-antiproton collisions, supporting the $L = 6\\sigma$ connection.","marker":"[40]"}],"fun_headline_variants":["Sound speed from hadron rapidity width: mostly ≤1/3","Rapidity spread of hadrons yields sound speed: mostly 0–1/3","Landau model + Gaussian width: c_s^2 mostly in 0 to 1/3","Sound speed in hot hadronic matter: mostly obeys conformal bound","Extract c_s^2 from Gaussian rapidity width: most ≤1/3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the spread of produced particles is exactly six times the rapidity gap between the leading nucleons after an assumed energy loss of 50 to 90 percent; neither the proportionality nor the loss window is measured in this paper.","fun_headline_variants_meta":{"raw":{"variants":["Sound speed from hadron rapidity width: mostly ≤1/3","Rapidity spread of hadrons yields sound speed: mostly 0–1/3","Landau model + Gaussian width: c_s^2 mostly in 0 to 1/3","Sound speed in hot hadronic matter: mostly obeys conformal bound","Extract c_s^2 from Gaussian rapidity width: most ≤1/3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000935,"raw_usage":{"total_tokens":4007,"prompt_tokens":961,"completion_tokens":3046,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":2937}},"tokens_in":577,"tokens_out":3046,"duration_ms":21317,"temperature":1.0,"reasoning_tokens":2937,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:17:03.798794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the full rapidity distribution in proton-proton collisions at, say, 200 GeV and 13 TeV, fit it to a single Gaussian, and compute $L$ from leading-nucleon energies using Eqs. (1)-(8). If the fitted $\\sigma/L$ differs from $1/6$ by more than experimental uncertainty, or if the $c_s^2$ extracted from Eq. (12) disagrees with the value obtained from lattice-QCD equation-of-state calculations under the same conditions, the central claim fails.","supporting_citations":[{"cited_title":"Viscosity, non- conformal equation of state and sound velocity in Landau hydrodynamics,","cited_arxiv_id":null,"evidence_quote":"Gives a non-conformal viscous Landau solution with an alternative sigma-squared/sound-speed relation and confirms slightly different rapidity widths."},{"cited_title":"On pseudorapidity distribution and speed of sound in high energy heavy ion collisions based on a new revised Landau hydrodynamic model,","cited_arxiv_id":null,"evidence_quote":"Previous application of the Landau-based extraction to Au-Au and Cu-Cu pseudorapidity distributions, giving $c_s^2$ near 1/3-1/2 for central and non-central regions."},{"cited_title":"On distributions of emission sources and speed of sound in proton-proton (proton- antiproton) collisions,","cited_arxiv_id":null,"evidence_quote":"Earlier extraction of $\\sigma$ and $c_s^2$ in proton-proton and proton-antiproton collisions, supporting the $L = 6\\sigma$ connection."}],"review_version":1}