{"id":"aaf79976-4302-49e5-a298-8f50d71595b2","arxiv_id":"2412.04994","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Excited cluster decays raise predicted proton cumulant ratios in a hadron resonance gas at collision energies of 2 to 5 GeV per nucleon pair, with the largest effect on high-order ratios at low energies.","lead":"Excited nuclear cluster decays are added to a statistical model of heavy-ion collisions, and their contribution to proton number fluctuation ratios is computed at collision energies between 2 and 5 GeV per nucleon pair. The effect grows toward low energies and can change the sixth-to-second cumulant ratio by about 100 percent, a baseline correction relevant for the CBM experiment at FAIR.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The at-most-one-proton assumption for excited clusters is contradicted by the cited decay tables (e.g., 6Be -> alpha + p + p), invalidating the binomial cumulant formulas that produce the headline corrections.","rationale":"The reader's weakest_assumption is exactly the load-bearing point. The strongest claim in the paper is quantitative: corrections from 1% to 100% in cumulant ratios from excited cluster decays. For those numbers to be trustworthy, the probability model for how many protons emerge from one cluster decay must be correct. Eq. (12) is a binomial compound model and is only valid when each cluster contributes at most one proton. The cited nuclear-data source [21] for A=5-7 contains well-known states with two-proton decay, of which 6Be(0+) -> alpha + p + p is the clearest example. Because the paper says it uses those tables for its excited clusters, the binomial premise is not merely an unverified approximation; it is plausibly inconsistent with the input data. A fix is straightforward: use the full decay-multiplicity distribution in the generating function. Until that check is done, the headline numbers, especially the sixth-order ratio, should be regarded as conditional. I do not see a reason to move the reader's verdict from CONDITIONAL: the model framework is reasonable, the formulas are otherwise internally consistent, and the concern is eminently testable rather than fatal. The appropriate action is to require the multi-proton audit and recomputation before the quantitative claim is taken as a baseline. The paper already includes an explicit assumption and cites the relevant tables, so this is a checkable correction, not a rejection. I also note the absence of uncertainty estimates, but that is secondary; the decay-multiplicity issue directly affects the central numbers.","tokens_in":7828,"tokens_out":7999,"duration_ms":184555,"concrete_test":"Audit the input decay list from Ref. [19] Tables I/II against the A=5-7 energy levels in Ref. [21], counting final-state protons for every included cluster. In particular, determine whether 6Be(0+) -> 4He + p + p is present. If it is, re-derive the cumulant-generating function for a cluster species with three outcomes (0, 1, or 2 protons), K_R(i*xi)=ln Sum_{N_R} P(N_R)(1-p_1-p_2+p_1 e^{i*xi}+p_2 e^{2i*xi})^{N_R}, feed the [19]/[21] branching ratios for 6Be and any other multi-proton states into Eqs. (12), and recompute Fig. 2 at sqrt(s_NN)=2.4 GeV. If kappa_H sigma^4 changes by more than a few percent relative to the binomial treatment, let alone the claimed ~100% correction, the central quantitative claim must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2 states: 'like for baryonic resonances, all included excited clusters produce at most one proton. Hence, the probability distribution ... is binomial.' This is the basis for the cumulant-generating function (11) and the explicit formulas (12a)-(12f). The input states are taken from Tables I/II of [19], which rely on the A=5-7 evaluation [21]. That evaluation lists the 6Be ground state (0+, T=1) in the A=6 table with its dominant strong decay channel 4He + p + p, i.e., two final-state protons. If this state (or any other A=5-7 state with a multi-proton channel) is included in the calculation, then p_R is not the probability of producing exactly one proton; the decay multiplicity is x=0,1,2, and the binomial factorial-cumulant structure of Eqs. (12) is invalid. The reported corrections, including the factor-of-two change in kappa_H sigma^4 at the lowest energy, would have to be recomputed with a multinomial/compound distribution. The paper neither excludes multi-proton channels nor justifies a selection that would remove them; since [19] and [21] are the cited sources, this is an internal mismatch rather than a model-choice objection.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper estimates the contribution of decays of excited nuclear cluster states to event-by-event proton number cumulant ratios in the statistical model, for sqrt(s_NN) between 2 and 5 GeV. The formalism treats each cluster as a resonance that produces at most one proton, uses the binomial cumulant formulas (12a)-(12f) from [23], and takes the cluster input from [19]. It reports corrections from about 1% for the second-to-first cumulant ratio to about 100% for the sixth-to-second cumulant ratio at the lowest energies, with the effect becoming negligible above 3.5 GeV. The authors argue that such corrections provide an improved baseline for CBM comparisons with lattice QCD and effective models.","tokens_in":8109,"tokens_out":5597,"duration_ms":61453,"significance":"If correct, this is a useful quantitative baseline improvement for proton cumulant measurements in the high-baryon-density regime. The calculation is non-circular: the freeze-out parameters come from an independent fit to hadron multiplicities [24], the cluster input comes from external nuclear data evaluations via [19], and no fluctuation observable is fitted. The cumulant recursion is transparent, the volume cancels in the reported ratios, and the final ratios constitute a falsifiable prediction for CBM. The main risk is technical rather than conceptual: the binomial treatment rests on an unverified at-most-one-proton assumption, and the p_R values and state selection are not documented. The significance is moderate: this is an incremental but potentially important baseline correction, not a new phenomenon or a new formalism.","major_comments":[{"comment":"The cumulant-generating function (11) and the explicit formulas (12a)-(12f) rely on the statement that 'all included excited clusters produce at most one proton.' The manuscript does not verify this against its adopted input: the A=5-7 evaluation [21] lists the 6Be ground state with a dominant alpha+p+p decay channel, and the text does not state that such states are excluded from Tables I/II of [19]. If any included state has a two-proton decay channel, p_R is not the probability of producing exactly one proton, the binomial factorial-cumulant structure of (12) is invalid for that species, and the reported corrections—including the roughly 100% change in kappa_H sigma^4 at low energy—must be recomputed with a compound or multinomial decay distribution. This is load-bearing because the central quantitative claim follows directly from these formulas.","section":"Sec. 2, Eqs. (11)-(12)"},{"comment":"The paper gives no explicit list of the excited cluster states included, no branching ratios, and no prescription for converting the decay tables in [19]/[21] into the p_R values entering Eqs. (12). Without this information the calculation is not reproducible, and one cannot assess how strongly the headline 100% correction depends on the inclusion or exclusion of specific states. The authors should provide the state list with adopted p_R values and state explicitly how multi-body decay channels are treated.","section":"Sec. 3, cluster input and p_R"},{"comment":"Eq. (14) assumes resonances and clusters have vanishing widths, yet the excited cluster states taken from [21] often have substantial widths. The finite-width effect on the mean and higher cumulants in Eqs. (15)-(16) is not estimated. Since the paper highlights corrections at the 1% to 100% level, a quantitative argument that finite widths change the reported ratios by less than, say, the claimed effect size is needed before the numbers can be considered robust.","section":"Sec. 2, Eq. (14)"}],"minor_comments":[{"comment":"For i=1 the definition r_i1 is undefined, since the denominator becomes 1 - <N>_c/<N>_c = 0; the text should specify that Eq. (19) applies for i>=2 or otherwise define the i=1 case.","section":"Sec. 3, Eq. (19)"},{"comment":"The caption states that the curves collapse to 'two curves' but does not identify which curve corresponds to ratios with <N>_c in the denominator and which to ratios with <(Delta N)^2>_c in the denominator; add a legend or explanatory note.","section":"Fig. 3"},{"comment":"The symbol 'NR' appears as 'the number of protons resulting from decays of NR resonances'; this should be typeset as N_R (resonance multiplicity) to avoid confusion with the resonance species label R.","section":"Sec. 2, text above Eq. (11)"},{"comment":"The terms 'hyperskewness' and 'hyperkurtosis' are not standard in heavy-ion literature; please define them or cite standard usage, as some readers will confuse the fifth and sixth standardized cumulants with skewness and kurtosis of hyperdistributions.","section":"Sec. 2, Eqs. (6)-(7)"},{"comment":"The phrase 'cuts up to one sixth from the distance of the cumulant ratio from unity' is unclear; please rephrase to state precisely what the scaling variable r_ij measures and what the 'one sixth' value implies quantitatively.","section":"Sec. 3, text after Fig. 3"},{"comment":"Reference [19] is incomplete: the volume/page is missing ('Phys. Lett. B (2020) 135746'); please provide the full citation.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope and the central idea is worth publishing if the technical issues are fixed. I would not reject on circularity grounds: the calculation is explicitly a baseline prediction using external inputs, which is appropriate. However, the at-most-one-proton assumption is not merely a presentation issue—it is the mathematical foundation of the reported 100% correction, and it is contradicted by at least one state in the cited evaluation. The authors should either demonstrate that all included states satisfy the assumption or recompute the cumulants with the correct decay multiplicity distribution. They should also provide the p_R table for reproducibility. After these changes, a moderate revision should be able to resolve the concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a legitimate new application of the HRG cumulant formalism to excited nuclear cluster feeddown, and it identifies a gap in baseline predictions for CBM. The qualitative message is plausible: excited cluster decays can shift high-order proton cumulant ratios at sqrt(sNN) below about 3.5 GeV. The quantitative claim, though, is built on an assumption that is contradicted by the paper's own cited decay tables.\n\nSection 2 asserts that all included excited clusters produce at most one proton, and that justifies the binomial treatment in Eqs. (12). The decay tables from [19] and [21] include 6Be, whose ground state decays dominantly to alpha + p + p. That is a two-proton channel. If 6Be or any other multi-proton state is in the input list, the binomial cumulant generating function (11) is invalid for those states, and the corrections—including the factor-of-two change in kappa_H sigma^4—need recomputation with a compound multinomial distribution. The paper neither checks this nor excludes such states.\n\nWhat is genuinely good: the derivation of (12) from (11) is internally consistent, the numerical procedure is transparent, the volume-cancellation point is handled cleanly, and the scaling observation in Fig. 3 is a nice way to present the effect. This is a real gap in the literature; previous feeddown calculations looked at yields, not cumulants.\n\nMinor but real issues: branching ratios and state lists are imported without reproduction, so a referee cannot independently confirm which states were included or how p_R was assigned. Decay widths are set to zero without discussion. There are no uncertainty estimates. These are secondary, but they make it harder to judge how robust the 1%–100% range is.\n\nBottom line: the paper deserves a serious referee. The fix is not hard—exclude multi-proton channels with justification, or switch to the proper compound distribution—but without it the headline numbers are not reliable. I would conditionally accept after that check, and ask the authors to supply the state list and p_R values as a table. The qualitative conclusion about excited cluster feeddown at low energies will likely survive the fix.","headline":"A useful new baseline for CBM-era proton cumulants, but the headline corrections rest on an unverified at-most-one-proton assumption that the cited decay tables likely contradict.","tokens_in":8604,"tokens_out":4578,"would_cite":false,"duration_ms":43790,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["25.75.-q","24.60.-k"],"model":"deepseek-v4-flash","headline":"Excited nuclear cluster decays shift proton number fluctuation ratios by up to 100 percent at low collision energies.","keywords":["proton number fluctuations","QCD phase diagram","relativistic heavy-ion collisions","excited nuclear clusters","hadron resonance gas","higher-order cumulants","CBM"],"falsifier":"Check the A=5–7 decay tables used in the paper for a state with a two-proton branch—6Be decaying to an alpha plus two protons would suffice—and if such a branch exists, recompute the sixth-to-second cumulant ratio with the true multinomial decay probabilities; the result would show whether the reported 100% correction survives.","tokens_in":7631,"feed_emoji":"⚛️","tokens_out":13796,"duration_ms":126949,"temperature":0.7,"pith_summary":"This paper identifies a previously neglected source of event-by-event proton number fluctuations in heavy-ion collisions: the decay products of excited nuclear cluster states. Using a statistical (hadron-resonance-gas) model, the authors include A=5–7 excited clusters as proton-feeding resonances and compute the first six cumulant ratios of the net-proton distribution for collision energies $\\sqrt{s_{NN}}$ from 2 to 5 GeV. They find that the clusters change every cumulant ratio, from about 1% for the variance-to-mean ratio to roughly 100% for the sixth-to-second ratio at the lowest energy. The correction is substantial below $\\sqrt{s_{NN}} \\approx 3.5$ GeV and falls away above about 5 GeV, moving each ratio closer to unity. The authors' point is that planned CBM measurements of high-order proton cumulants can be compared with lattice QCD and effective-model predictions only if this feeddown is included in the baseline.","feed_headline":"Excited cluster decays double proton fluctuation ratio at low energies","feed_subtitle":"Below 3.5 GeV, standard hadron-gas baselines miss a feeddown that shifts high-order proton cumulant ratios.","key_machinery":"The binomial feeddown cumulant machinery: for each resonance species $R$, the proton-number cumulant-generating function is built from the probability $p_R$ that a decay produces one proton, with direct protons treated as $p_R = 1$. Derivatives of this function give the first six proton cumulants as sums over all resonance species—including the excited clusters—of terms in $p_R$ and the species' grand-canonical number cumulants. The excited clusters enter through the same binomial formulas, and their decay probabilities and masses are taken from the imported light-nucleus tables. Because the cumulant ratios are volume-independent, the calculation needs only the chemical freeze-out temperature $T$ and baryochemical potential $\\mu_B$, which are fixed by a parametrization of $\\sqrt{s_{NN}}$; this machinery is what converts the decay tables into the reported corrections.","core_discovery":"The paper claims that the standard hadron-resonance-gas baseline for proton cumulant ratios has been missing an entire class of sources: the decay products of excited nuclear cluster states with A=5–7. Treating these clusters exactly like resonances that decay into at most one proton, and using the binomial cumulant formulas derived from the cumulant-generating function, the authors calculate the first six cumulant ratios of the (net-)proton number at chemical freeze-out. Including the clusters raises every ratio toward unity; at the lowest inspected energy the sixth-to-second cumulant ratio nearly doubles, while the variance-to-mean ratio moves by only about one percent. The effect is visible for $\\sqrt{s_{NN}}$ below about 3.5 GeV and becomes negligible above about 5 GeV. The authors frame the result as a necessary improvement to baseline calculations for CBM.","pith_inferences":["The reported size of the correction is sensitive to the assumption that every excited cluster emits at most one proton; if the imported decay tables contain a two-proton channel such as 6Be decaying to an alpha plus two protons, the binomial formulas would need to be replaced by multinomial ones and the 100% sixth-order effect could change.","Because all the $r_{ij}$ curves collapse onto two universal curves, one could parametrize the cluster correction for any cumulant ratio as a function of (1 minus the ratio) and apply it to existing hadron-resonance-gas calculations without recomputing the full thermal model.","The same feeddown logic should carry over to other conserved-charge cumulants, so excited cluster decays may also affect net-baryon, electric-charge, or strangeness fluctuation ratios at low beam energies—an extension the paper does not pursue.","A direct data-level test: if high-order proton cumulant ratios in central Au+Au at the lowest CBM energies rise toward unity in a way standard hadron-resonance-gas models do not reproduce, that rise would match the cluster-feeddown mechanism described here."],"forward_implications":["At $\\sqrt{s_{NN}}$ below about 3.5 GeV, hadron-resonance-gas baselines that omit excited cluster feeddown understate the sixth-to-second-order proton cumulant ratio by up to a factor of two.","All the cumulant ratios examined move closer to unity when excited clusters are included, with the higher-order ratios affected most strongly.","Above about 5 GeV the correction is negligible, so high-energy fluctuation baselines do not need to be changed.","The scaling variable $r_{ij}$ collapses the corrected ratios onto two energy-dependent curves, meaning the correction can be summarized as a single factor applied to the distance of each ratio from unity—up to one sixth at 2.4 GeV.","Direct comparisons of CBM proton-fluctuation data with lattice QCD or effective-model results at FAIR energies require this feeddown to be part of the model baseline."],"supporting_citations":[{"why":"supplies the list of excited nuclear cluster states and their decay probabilities that are added to the resonance gas","marker":"[19]"},{"why":"provides the A=4 light-nucleus energy-level data from which cluster states are taken","marker":"[20]"},{"why":"provides the A=5, A=6, A=7 energy-level tables that define the excited clusters and their decay channels","marker":"[21]"},{"why":"gives the hadron-resonance-gas treatment of resonance decay and regeneration effects on net-proton fluctuations","marker":"[22]"},{"why":"supplies the cumulant-generating function and the binomial formulas (Eqs. 11 and 12) that the paper extends to excited clusters","marker":"[23]"},{"why":"provides the chemical freeze-out parametrization of $T$ and $\\mu_B$ as functions of collision energy used in the calculation","marker":"[24]"}],"fun_headline_variants":["Excited clusters double proton cumulant ratio at low energies","Missing cluster feeddown shifts proton fluctuations at CBM energies","Proton cumulant ratios need excited nuclear clusters below 3.5 GeV","New proton fluctuation source from excited cluster decays"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes that every excited nuclear cluster included in the decay tables produces at most one proton; the paper does not verify this channel-by-channel, and if any included A=5–7 state has a two-proton decay branch the binomial cumulant formulas and the reported correction sizes would need to be recomputed.","fun_headline_variants_meta":{"raw":{"variants":["Excited clusters double proton cumulant ratio at low energies","Missing cluster feeddown shifts proton fluctuations at CBM energies","Proton cumulant ratios need excited nuclear clusters below 3.5 GeV","New proton fluctuation source from excited cluster decays"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000151,"raw_usage":{"total_tokens":1162,"prompt_tokens":870,"completion_tokens":292,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":224}},"tokens_in":486,"tokens_out":292,"duration_ms":3964,"temperature":1.0,"reasoning_tokens":224,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:01:52.495566+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the A=5–7 decay tables used in the paper for a state with a two-proton branch—6Be decaying to an alpha plus two protons would suffice—and if such a branch exists, recompute the sixth-to-second cumulant ratio with the true multinomial decay probabilities; the result would show whether the reported 100% correction survives.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the A=4 light-nucleus energy-level data from which cluster states are taken"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the A=5, A=6, A=7 energy-level tables that define the excited clusters and their decay channels"},{"cited_title":"Impact of resonance regeneration and decay on the net-proton fluctuations in a hadron resonance gas","cited_arxiv_id":"1402.1238","evidence_quote":"gives the hadron-resonance-gas treatment of resonance decay and regeneration effects on net-proton fluctuations"}],"review_version":1}