{"id":"2837cad6-6c0d-4d98-bdc4-ea0fca388509","arxiv_id":"2507.04809","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":8,"one_line_summary":"Decoherence of a glasma-like coherent state yields an entropy per particle below the thermal gluon gas value, except in proton-nucleus collisions at small g mu.","lead":"This paper calculates how much entropy is created when the coherent gluon field state formed just after a heavy-ion collision loses quantum coherence against vacuum fluctuations. It finds that the entropy per particle falls below the value of a thermalized gluon gas, so decoherence by itself cannot explain thermalization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"S_infinity/N depends on the unspecified longitudinal box length Lz, so the numerical comparison with the thermal value is underdetermined and the central claim does not follow from Eqs. (86)-(93).","rationale":"Good faith: the paper wants to determine whether phase-damping decoherence of a glasma-like coherent state can produce entropy per particle comparable to a thermalized gluon gas. The single-mode decoherence formula (28) is standard, and the MV-model occupation-number calculation is internally organized. But the central object S_infinity/N is not invariant under the quantization-box choice. Because the authors keep only kz = 0 modes and define n_k through Eq. (45), the combination d3k brings one power of Lz while <|a|^2> carries Lz^2; the net n_k is linear in Lz. In converting sums to integrals, Eq. (91) contains only (2pi)^2/AT, so no Lz is removed. The Poisson entropy is nonlinear, so S_infinity/N genuinely depends on Lz. This is the first half of the reader's weakest_assumption, and I agree it is load-bearing. The second assumption, replacing event-averaged occupations by a single coherent amplitude per mode, is also real; since S_infinity(n) is concave, the plotted S_infinity/N is an upper bound to the event-averaged value. That would strengthen the main 'insufficient entropy' conclusion for most cases but would weaken the claimed pA exception. The Lz issue is more fundamental because it can move S_infinity/N both above and below the thermal line depending on the arbitrary choice, so the quantitative claim as written does not follow. The paper provides no machine-checked proof or reproducible code, and the listed parameters (m, AT, c, xg) do not fix Lz. Therefore the reader's REJECT verdict is appropriate and I would leave it unchanged.","tokens_in":17766,"tokens_out":13769,"duration_ms":171479,"concrete_test":"Recompute the AA curve in Fig. 5 at fixed gmu = 2 GeV, m = 0.2 GeV, AT = 9 fm^2 using Eq. (86) with Lz = 1/Qs (about 0.2 fm) and with Lz = 1 fm. If S_infinity/N changes by more than 20% between these choices, the headline comparison is an artifact of an unspecified parameter. Additionally, redo the derivation with a finite longitudinal rapidity extent Delta_eta and verify whether the Lz dependence cancels; if it does not, the central ratio is not well defined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eqs. (86)-(87) give n_k proportional to Lz (AA) and Lz/AT (pA), while the mode sums in Eqs. (92)-(93) are only over transverse momenta with dkx dky = (2pi)^2/AT; there is no compensating dkz or 1/Lz factor. Thus the argument of the Poisson entropy S_infinity(n_k) = -sum_l e^{-n_k} n_k^l/l! log(...) contains the never-specified Lz. Since S_infinity(n) is nonlinear (about n(1-log n) for n<<1 and about 1/2 log(2pi e n) for n>>1), N = sum n_k and S_infinity = sum S_infinity(n_k) scale differently with Lz, so S_infinity/N is a function of the arbitrary Lz. The paper never states a value or physical justification for Lz, so Figs. 4-5 and the comparison to sSB = 2.19 are not determined by the equations. A second independent issue is the replacement beta_k = sqrt(<|a_k|^2>) in Eq. (46): because S_infinity is concave, the event-averaged entropy is overestimated. However, the Lz ambiguity alone is sufficient to invalidate the quantitative claim as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the asymptotic von Neumann entropy produced by phase-damping decoherence of a coherent state whose mode occupation numbers are taken from the McLerran-Venugopalan Glasma in AA and pA collisions. It compares the decoherence entropy per particle S∞/N with the entropy per particle of a thermal ultrarelativistic gluon gas and concludes that decoherence alone is insufficient to thermalize, except possibly for pA at small gμ and large transverse areas.","tokens_in":18073,"tokens_out":6326,"duration_ms":67798,"significance":"If fully established, the result would be a useful step in quantifying the role of decoherence in the early-stage thermalization problem. The single-mode decoherence result (Eqs. 23–28) is a standard and correct application of the phase-damping model, and the MV-model computation of the occupation numbers (Secs. IV) is internally consistent and clearly presented. The paper explicitly identifies the limitations of the phase-damping model. However, the central quantitative comparison is compromised by two load-bearing issues.","major_comments":[{"comment":"The ratio S∞/N is not well defined because the occupation numbers scale with the longitudinal box length Lz. For AA, n_k ∝ Lz (Eq. 86); for pA, n_k ∝ Lz/AT (Eq. 87). In the mode sums of Eqs. (92)–(93), the integration is only over transverse momenta with dkx dky = (2π)^2/AT; there is no compensating dkz or factor 1/Lz. Since the Poisson entropy S∞(n_k) is a nonlinear function of n_k (Eq. 28), S∞ and N scale differently with Lz, so S∞/N depends on the arbitrary, never-specified Lz. The paper never states a value or physical justification for Lz, and Figs. 4–5 therefore do not follow from the equations. The central comparison with the thermal value sSB = 2.19 is underdetermined.","section":"Eqs. (69), (86)–(93), Figs. 4–5"},{"comment":"The replacement of the event-averaged Glasma by a single coherent state with amplitude β_k = sqrt(⟨|a_k|^2⟩) is not justified. The MV model produces a distribution of color-charge configurations and hence a distribution of occupation numbers n_k per event. The decoherence entropy S(n_k) is a concave function of n_k, so by Jensen's inequality S(⟨n_k⟩) ≥ ⟨S(n_k)⟩. The calculation therefore overestimates the event-averaged entropy, and the reported S∞/N is not the entropy per particle of the decohered Glasma ensemble. The paper neither computes the distribution of n_k nor quantifies the resulting error.","section":"Eq. (46), Sec. III.A"}],"minor_comments":[{"comment":"The notation for the coherent-state amplitude changes from α (Sec. II) to β (Sec. III) without explanation; this is confusing when comparing Eqs. (10) and (40).","section":"Secs. II and III"},{"comment":"The numerical value of Lz used for the curves is not stated anywhere; the captions and text should specify all inputs needed to reproduce the results.","section":"Figs. 4–5"},{"comment":"The abstract's qualifier 'except for proton-nucleus collisions at small values of gμ' is not precise, since the small-gμ range depends on the unstated Lz and on AT; the statement should be reformulated after the Lz issue is resolved.","section":"Abstract"}],"recommendation":"reject","confidential_remarks":"The Lz dependence alone invalidates the central quantitative claim as it currently stands, and the ensemble-averaging issue compounds the problem. If the authors can supply a physically motivated Lz (e.g., from the longitudinal extent of the Glasma slice) and redo the comparison, and also properly average the decoherence entropy over the event distribution of occupation numbers, a resubmission could be considered. As written, the paper's main conclusion is not supported by the presented calculation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper is a straightforward, honest application of textbook phase-damping decoherence to coherent states built from MV-model Glasma occupation numbers; the algebra is internally consistent and the AA/pA distinction is a useful addition. Second, the headline claim—that S/N after decoherence sits below the 2D thermal value—does not follow from the equations as written, because the result depends on the never-specified longitudinal box length Lz.\n\nWhat is genuinely new: the MV-model computation of n_k for coordinate-independent (AA) and hotspot (pA) color charges, and the quantitative comparison of the decohered S/N with the 2D Bose gas value. The single-mode decoherence formula is standard, and the Glasma-to-coherent-state mapping was already in Ref. [47], but the specific scan over gμ and the AA vs pA contrast is new territory.\n\nThe soft spot is load-bearing. Eq. (86) has n_k ∝ Lz for AA, and Eq. (83) has n_k ∝ Lz/AT for pA. N is linear in n_k, but the Poisson entropy S∞(n_k) is nonlinear (about n(1-log n) for small n, about 1/2 log(2πen) for large n). So when you scale Lz, N and S∞ scale differently and S∞/N changes. The paper never states a value or physical motivation for Lz. I checked whether the transverse-mode measure cancels it: it doesn't, because the sum is only over kx,ky with dkxdky = (2π)^2/AT, and there is no compensating 1/Lz. This is not a minor issue; Figs. 4–5 and the comparison with sSB=2.19 are literally functions of an unspecified parameter. The event-averaging replacement β_k = sqrt(⟨|a|^2⟩) is a second weakness—it overestimates the entropy because S(n) is concave—but the Lz ambiguity alone is enough to invalidate the quantitative claim as stated.\n\nIs the paper worth engaging with? Yes, as a serious referee. The derivation is clear, the model is physical, and the Lz problem may be fixable by choosing Lz ~ 1/Qs (or whatever the relevant longitudinal extent is) and showing sensitivity. The qualitative direction—phase decoherence alone underproduces entropy relative to a thermal gluon bath—is plausible and consistent with the literature, but the numbers in Figs. 4–5 are not yet determined. I'd ask the authors to specify Lz, justify it, and rerun the plots; if the conclusion survives, the paper becomes a useful reference for pre-equilibrium modelers. As it stands, I wouldn't cite the quantitative claim.","headline":"A clean application of phase-damping decoherence to Glasma occupation numbers whose central quantitative claim is underdetermined by an unspecified longitudinal box length Lz.","tokens_in":18654,"tokens_out":10316,"would_cite":false,"duration_ms":107446,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Aw","12.38.Mh"],"model":"deepseek-v4-flash","headline":"The entropy per particle produced by decoherence of the glasma coherent state falls below the two-dimensional thermal gluon-gas value, except for proton-nucleus collisions at small coupling.","keywords":["decoherence","entropy production","open quantum systems","Glasma","coherent state","phase damping","von Neumann entropy","color glass condensate"],"falsifier":"Compute $S_\\infty/N$ using the full event distribution of $|a_k|^2$ instead of the single-coherent-state replacement $\\beta_k=\\sqrt{\\langle |a_k|^2\\rangle}$, and scan the longitudinal length $L_z$ in the occupation-number formulas: the central conclusion fails if the ratio crosses the thermal value $2.19$ or moves by more than a few percent as $L_z$ varies.","tokens_in":17514,"feed_emoji":"⚛️","tokens_out":12273,"duration_ms":120463,"temperature":0.7,"pith_summary":"The paper asks whether quantum decoherence alone can create enough entropy to turn the coherent gluon state formed in the earliest moments of a high-energy nuclear collision into a thermalized gluon bath. It builds a coherent state whose occupation numbers are taken from the glasma fields of proton-nucleus and nucleus-nucleus collisions, couples that state to the vacuum through a phase-damping interaction that preserves particle number, and computes the asymptotic von Neumann entropy per particle $S_\\infty/N$. The answer it finds is mostly no: apart from proton-nucleus collisions at small values of the coupling parameter $g\\mu$, the decohered entropy per particle stays below the value $\\simeq 2.19$ of a two-dimensional ultrarelativistic Bose gas of gluons. For nucleus-nucleus collisions at LHC-like values the ratio is only about one tenth of the thermal value. The paper concludes that phase decoherence from vacuum fluctuations is insufficient for thermalization, so additional entropy-generating mechanisms must operate.","feed_headline":"Decoherence alone produces too little entropy to thermalize the glasma","feed_subtitle":"Even in the asymptotic limit, entropy per gluon stays below a 2D thermal gas except for small proton-nucleus systems.","key_machinery":"The central object is the phase-damping model of decoherence, defined by the system-reservoir Hamiltonian $H_{SR}=a^\\dagger a(\\Gamma^\\dagger+\\Gamma)$ and the vacuum Lindblad master equation. This interaction kills the off-diagonal elements of the density matrix exponentially while leaving occupation numbers intact, so each mode relaxes to the Poisson mixture $\\rho_\\infty = \\sum_\\ell e^{-n_k} n_k^\\ell/\\ell!\\,|\\ell\\rangle\\langle\\ell|$. The map from glasma fields to occupation numbers is carried by the standard color-glass-condensate color-charge correlators, yielding $n_k^{AA}\\propto L_z (g\\mu_A)^2(g\\mu_B)^2 I(k_T)/\\omega_k$ and the analogous $pA$ expression, where $I(k_T)$ is a UV- and IR-finite transverse momentum integral. The comparison target is $S_{\\rm SB}/N = 2.19$ for a two-dimensional ultrarelativistic Bose gas of gluons.","core_discovery":"The central discovery is that the asymptotic entropy of a decohered glasma coherent state is a sum over modes of the Shannon entropy of the Poisson distribution $\\lambda_\\ell = e^{-n_k} n_k^\\ell/\\ell!$, and that when the $n_k$ are taken from the glasma occupation numbers this entropy per particle falls short of the thermal benchmark. For nucleus-nucleus collisions, $S_\\infty/N$ is independent of the transverse area and stays well below the two-dimensional ultrarelativistic gluon gas value $s_{SB}=2.19$; for proton-nucleus collisions it lies in the range roughly $0.35$--$0.75$ of $s_{SB}$ at LHC-relevant couplings, reaching or exceeding the thermal value only for the smallest $g\\mu$ and largest transverse areas. As $g\\mu$ grows the initial coherent state becomes denser and $S_\\infty/N$ falls further behind the thermal value, because the total occupation number $N$ grows faster than the entropy $S_\\infty$.","pith_inferences":["Editorial extension: replacing the single coherent amplitude $\\beta_k=\\sqrt{\\langle |a_k|^2\\rangle}$ with the full distribution of $|a_k|^2$ would shift $S_\\infty/N$, since entropy is nonlinear in $n_k$; a direct numerical test would settle the direction and size of the shift.","Editorial extension: because the environment is taken to be the vacuum, the computed entropy is a lower bound; coupling the glasma to a thermal reservoir through amplitude damping should add both particles and entropy, and the model predicts the $S/N$ gap would close as the reservoir temperature rises.","Editorial extension: the same Poisson-entropy map could be applied to anisotropic or overoccupied pre-equilibrium distributions to test whether subthermal decoherence entropy is generic for saturated gluon states."],"forward_implications":["For nucleus-nucleus collisions at $g\\mu \\simeq 2$ GeV, the decohered state reaches only about ten percent of the two-dimensional thermal gluon-gas entropy per particle, so most of the final entropy must come from later non-abelian dynamics.","As the glasma density parameter $g\\mu$ increases, $S_\\infty/N$ moves further below the thermal benchmark because $N$ grows faster than $S_\\infty$; denser initial coherent states are therefore farther from decoherence-driven thermalization.","The two-dimensional thermal benchmark $S_{\\rm SB}/N = 2.19$ is the appropriate comparison for a boost-invariant glasma; using the three-dimensional value $3.60$ would make the shortfall even larger.","In proton-nucleus collisions, decoherence brings the system closer to thermal equilibrium than in nucleus-nucleus collisions, with the gap closing only at small $g\\mu$ and large transverse area."],"supporting_citations":[{"why":"Provides the glasma-to-coherent-state mapping and the normal-mode ladder-operator assumption the paper adopts.","marker":"[47]"},{"why":"Introduces the color-glass-condensate color-charge correlator from which the occupation numbers are derived.","marker":"[29]"},{"why":"Supplies the analytic phase-damping solution for the density matrix that gives the Poisson asymptotic state.","marker":"[54]"},{"why":"Underlies the master-equation and Lindblad derivation that defines the decoherence dynamics.","marker":"[23]"},{"why":"Fixes the hot-spot proton transverse profile used for the proton-nucleus occupation numbers.","marker":"[55]"},{"why":"Supplies the decoherence timescale estimate that frames the asymptotic entropy as physically relevant.","marker":"[52]"},{"why":"Sets the physically relevant range of $g\\mu$ used in the LHC-oriented conclusions.","marker":"[58]"}],"fun_headline_variants":["Decoherence entropy too low to thermalize glasma","Glasma decoherence falls short of thermal entropy","Entropy from decoherence: below thermal benchmark","Decoherence fails to thermalize glasma states","Glasma entropy from decoherence: sub-thermal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the event-averaged glasma can be replaced, mode by mode, by one coherent state of amplitude $\\beta_k=\\sqrt{\\langle |a_k|^2\\rangle}$, and that the longitudinal box length $L_z$ cancels from the entropy per particle; because the von Neumann entropy is nonlinear in the occupation number, neither step is guaranteed, and $L_z$ is never fixed in the paper.","fun_headline_variants_meta":{"raw":{"variants":["Decoherence entropy too low to thermalize glasma","Glasma decoherence falls short of thermal entropy","Entropy from decoherence: below thermal benchmark","Decoherence fails to thermalize glasma states","Glasma entropy from decoherence: sub-thermal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000683,"raw_usage":{"total_tokens":3102,"prompt_tokens":949,"completion_tokens":2153,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":2073}},"tokens_in":565,"tokens_out":2153,"duration_ms":16210,"temperature":1.0,"reasoning_tokens":2073,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:41:45.254512+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $S_\\infty/N$ using the full event distribution of $|a_k|^2$ instead of the single-coherent-state replacement $\\beta_k=\\sqrt{\\langle |a_k|^2\\rangle}$, and scan the longitudinal length $L_z$ in the occupation-number formulas: the central conclusion fails if the ratio crosses the thermal value $2.19$ or moves by more than a few percent as $L_z$ varies.","supporting_citations":[{"cited_title":"Diffusion of heavy quarks in the early stage of high-energy nuclear collisions at energies available at the BNL Relativistic Heavy Ion Collider and at the CERN Large Hadron Collider","cited_arxiv_id":"1911.02480","evidence_quote":"Provides the glasma-to-coherent-state mapping and the normal-mode ladder-operator assumption the paper adopts."},{"cited_title":"Entropy production from chaoticity in Yang-Mills field theory with use of the Husimi function","cited_arxiv_id":"1603.04622","evidence_quote":"Supplies the analytic phase-damping solution for the density matrix that gives the Poisson asymptotic state."},{"cited_title":"Entropy production in classical Yang-Mills theory from Glasma initial conditions","cited_arxiv_id":"1304.1807","evidence_quote":"Fixes the hot-spot proton transverse profile used for the proton-nucleus occupation numbers."},{"cited_title":"Entropy production in longitudinally expanding Yang-Mills field with use of Husimi function$-$semiclassical approximation","cited_arxiv_id":"2203.02859","evidence_quote":"Supplies the decoherence timescale estimate that frames the asymptotic entropy as physically relevant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets the physically relevant range of $g\\mu$ used in the LHC-oriented conclusions."}],"review_version":1}