{"id":"b15096d2-964a-41ab-a739-2de921d0cc19","arxiv_id":"2501.00548","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Finite-volume momentum-space constraints in a mean-field quark-meson model shift the critical endpoint significantly for L<10 fm and displace baryon fluctuation signals, with the shift direction depending on the chosen scenario.","lead":"This paper computes how a finite system size shifts the predicted critical endpoint of the QCD phase diagram within a mean-field quark-meson model, using three different ways to restrict quark momenta. It finds shifts below L≈10 fm that depend strongly on the chosen momentum-space constraint, which matters for interpreting heavy-ion collision data in the search for the critical endpoint.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Vacuum-modified discretization scenarios are excluded by an unbounded grand potential, so the advertised 'in most cases' CEP shift below L≈10 fm rests on model subsets; the fluctuation claim is shown for a single model at L=6 fm.","rationale":"The reader identified the modeling assumption (momentum-space constraints as a proxy for finite volume) as the load-bearing premise; I agree with that concern and with the CONDITIONAL verdict. However, I think the sharper internal issue is the author's own exclusion of the most complete discretized-vacuum scenarios because of the unbounded grand potential, and the resulting scenario-dependence that undermines the headline generality ('in most cases' and 'shape hardly modified'). The reader lists this as secondary limitation (ii); I elevate it to the primary stress point while keeping the verdict CONDITIONAL, since the concern is about scope and representativeness rather than an identified computational error.","tokens_in":9504,"tokens_out":1577,"duration_ms":13832,"concrete_test":"Fix one model (e.g., ePQM Fit1,1,1,2). Compute the L-dependence of the CEP with the vacuum-fluctuation discretization implemented with a controlled renormalization that restores boundedness (e.g., a Pauli-Villars or subtractive scheme, or evaluating the vacuum sum at finite UV cutoff with counterterms fit to L=∞). If the CEP path then differs qualitatively from the excluded case, the 'in most cases' claim must be restricted. Separately, recompute Fig. 3 cumulant ratios at L=8 fm and L=5 fm for both APBC discretization and low-momentum cutoff; if the peak shift direction or height change reverses between L values, the 'shape hardly modified' statement fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Sec. 5) that the CEP is significantly shifted below L≈10 fm in 'most cases' depends on which momentum-space scenario is counted. In Sec. 3 the author states a self-admitted obstruction: when vacuum fluctuations are modified by discretization, the grand potential is not bounded from below as φ̄→∞ (∝ −φ̄^4 log φ̄ term), so no common solution of the field equations exists for ePQM around L∼5.5 fm. The paper therefore restricts the phase-diagram study to (i) the low-momentum cutoff, (ii) APBC/PBC with unmodified vacuum, and (iii) PBC with only the zero mode discretized in vacuum. These cases are arguably the least complete finite-size implementations: a cutoff cannot arise from a spatial box, and leaving the vacuum continuum modes unmodified while discretizing matter modes is internally inconsistent with the mode-sum rationale of Sec. 2.1. Meanwhile, the modified-vacuum low-momentum-cutoff case makes the CEP disappear already at L≈2.5 fm, so the generic statement is not universal even within the paper's own scenarios. For the baryon-fluctuation claim (Sec. 4), the result is computed for one detached model (Polyakov-extended QM B, m_σ=600 MeV, no vacuum fluctuations) at a single size L=6 fm, with the disclaimer that multiple critical points and staircase transitions are avoided by restricting to μ_q<250 MeV. The conclusion that 'the signal of the CEP is shifted, but its shape is hardly modified' therefore has narrow demonstrated support. The load-bearing assumption is not that finite-size effects vanish, but that the chosen scenarios are representative of genuine finite-volume physics; the paper itself shows strong scenario dependence, so the summary claim is not robustly supported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the finite-size dependence of the QCD phase diagram and baryon number fluctuations in a mean-field quark-meson model by imposing momentum-space constraints: a low-momentum cutoff and mode discretization with periodic (PBC) and antiperiodic (APBC) boundary conditions, with the UV-improved summation taken from Ref. [30]. It reports that, depending on the scenario and on whether vacuum fluctuations are modified, the critical endpoint (CEP) is significantly shifted for L < 10 fm, and that the shape of the cumulant-ratio signals along the phase boundary is hardly modified (shown for one model at L=6 fm). The paper concludes that finite-size effects must be included in comparisons between effective-model phase diagrams and heavy-ion data.","tokens_in":9849,"tokens_out":6329,"duration_ms":59059,"significance":"If the results hold, the paper provides a useful mapping of how different momentum-space finite-size implementations affect the CEP location and the baryon fluctuation ratios in a commonly used mean-field framework. The comparison across scenarios is informative and highlights that the choice of implementation is consequential. However, the paper's own analysis shows that some scenarios are excluded by an unbounded grand potential, and the fluctuation claim is demonstrated on a single parameter set at a single size; therefore the general statements in the abstract and conclusion are stronger than the evidence. The paper is a short proceedings contribution that relies heavily on Ref. [30] for technicalities, which is acceptable for the format but limits self-containedness.","major_comments":[{"comment":"The abstract claims the CEP is 'significantly shifted in each case' for L < 10 fm, but Sec. 5 and the body state 'in most cases'. In the modified-vacuum low-momentum-cutoff case (Fig. 1 top), the CEP disappears at L ≈ 2.5 fm and the chirally broken phase at L ≈ 2 fm, rather than being shifted. The wording should be made consistent, and the conclusion should be restricted to the cases where the CEP persists.","section":"Abstract and Sec. 3, Fig. 1"},{"comment":"The paper excludes the discretized-vacuum scenarios because the grand potential is unbounded from below when vacuum fluctuations are modified (leading to a -φ^4 log φ term), making the field equations have no common solution around L~5.5 fm for the ePQM. The remaining studied cases – unmodified vacuum, zero-mode-only vacuum discretization, and the low-momentum cutoff – are exactly those that are least directly tied to a finite spatial box. The low-momentum cutoff is not derivable from a finite volume, and leaving the vacuum continuum modes unmodified while discretizing matter modes is internally inconsistent with the mode-sum rationale of Eq. (2). Since the central claim of a generic CEP shift below L≈10 fm rests on these subsets, the paper should either justify why these scenarios are representative for heavy-ion fireballs or present the results as scenario-dependent rather than as a general statement.","section":"Sec. 3, unboundedness paragraph"},{"comment":"The claim that the CEP signal is 'shifted, but its shape is hardly modified' is supported by a single model (Polyakov-extended QM B with m_sigma=600 MeV, no vacuum fluctuations) at a single size L=6 fm, with mu_q restricted below 250 MeV to avoid multiple critical points. The figure shows that the peak in C4/C2 is shifted and the finite-size curves are not simply rescaled (e.g., the APBC curve exhibits non-monotonic structures). With no variation in L and no systematic check over parameter sets, the generality of the 'hardly modified' statement is not established. A range of sizes or a quantitative measure of the shape change is needed.","section":"Sec. 4, Fig. 3"},{"comment":"The paper states that the present results are 'either not complete ... or have too low resolution to see the scaling behavior near the CEP.' This self-admitted limitation bears directly on the fluctuation-shape conclusion: without a complete treatment of the finite-size divergences, the statement that the shape of the critical signal is 'hardly modified' should be treated as provisional. The authors should either provide a quantitative estimate of the uncertainty or soften the conclusion.","section":"Sec. 5, last paragraph"}],"minor_comments":[{"comment":"Footnote 1: 'Fruthermore' should be 'Furthermore'.","section":"Footnote 1"},{"comment":"'is directly applicable' should be 'are directly applicable' because the subject is 'momentum integrals' (plural).","section":"Sec. 2.1, after Eq. (1)"},{"comment":"The multiplicity sum over m is not defined; specify that m runs over all integer triplets with the same |p|.","section":"Sec. 2.1, Eq. (2)"},{"comment":"The dashed-dotted line is not described in the caption; indicate which scenario it corresponds to.","section":"Fig. 2 bottom panel caption"},{"comment":"The pressure is defined as Ω(0,0) − Ω(T,μ_q), but the normalization should be stated (e.g., grand potential density) to avoid ambiguity.","section":"Sec. 4, Eq. (7)"},{"comment":"In the caption of Figure 3, 'C4/C2 (T ≈ Tpc)' should specify that T is slightly below the transition temperature T_pc(μ_q) as stated in the text.","section":"Fig. 3 caption"},{"comment":"Reference [45] is incomplete; it should provide a journal or arXiv identifier.","section":"References"},{"comment":"The acronym ePQM is used without definition; define it at first use.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a proceedings-style contribution and relies heavily on Ref. [30] for technical implementation. The main concern is the mismatch between the abstract's 'in each case' claim and the more cautious body/conclusion, and the narrow support for the fluctuation-shape statement. If the journal expects a self-contained regular article, the amount of deferred technical detail may also be an issue. The paper is honest about its limitations, which is commendable, but the conclusions should be aligned with the demonstrated evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: a small, honest proceedings paper that extends your earlier work on momentum-space finite-size constraints in mean-field quark-meson models. The genuinely new pieces are the L-scan of the CEP path across four model variants (QM A/B/C and ePQM) for both PBC and APBC, the modified-vacuum low-cutoff comparison, and the L=6 fm cumulant ratios along the phase boundary. The figures support the qualitative statements in the text.\n\nWhat the paper does well: it does not oversell the framework. Section 3 openly says that if you discretize the vacuum contribution, the grand potential is unbounded below for phi→infinity (–phi^4 log phi), so no common solution exists for ePQM around L~5.5 fm. The author therefore restricts the phase-diagram study to scenarios that have a solution. That is a real limitation, and it is acknowledged rather than hidden. The calculations follow published mean-field equations and parameters (Schaefer–Wagner, Kovacs–Szep–Wolf, Lo et al.), so there is no circular fitting.\n\nNow the soft spots. The central claim in Sec. 5 – CEP significantly shifted below L≈10 fm \"in most cases\" – is more fragile than the abstract suggests. The cases that are actually solvable are exactly the least complete finite-size implementations: a low-momentum cutoff is not a box, and leaving vacuum continuum modes unmodified while discretizing matter modes is internally inconsistent with the mode-sum logic in Sec. 2.1. And the paper itself shows strong scenario dependence: with the modified-vacuum low-cutoff, the CEP disappears already at L≈2.5 fm. So \"in most cases\" really means \"in the subset of scenarios that are numerically well-defined,\" and the conclusions should be worded conditionally. The stress-test note lands on this, and I think it holds up.\n\nThe baryon-fluctuation part is thinner: one model (Polyakov-extended QM B, m_sigma=600 MeV, no vacuum fluctuations), one size (L=6 fm), and mu_q<250 MeV to dodge the staircase/multi-critical-point complications. The statement that the CEP signal is shifted but its shape is hardly modified is accurate for that case, but the demonstrated support is narrow.\n\nNo code or data are provided, which is normal for a proceedings contribution but does limit independent checking of the UV-improved summation.\n\nWho this is for: people working on effective-model finite-size corrections in the CEP search. It is a useful data point, not a decisive one.\n\nMy take: it deserves serious peer review. The computation is real and the author handles the main obstruction honestly. I would accept it, but I would ask the author to tighten the conclusion so it does not ride \"in most cases\" when several scenarios are excluded or move the other way. For my own work, I wouldn't cite it in the next 12 months; the model is too far from my questions. Reading group: maybe.","headline":"Finite-size CEP shifts in mean-field QM models: plausible within the model, but the advertised generality is undermined by the paper's own scenario dependence.","tokens_in":10447,"tokens_out":2823,"would_cite":false,"duration_ms":24895,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81V05","82B26"],"pacs":["12.38.Mh","25.75.Nq","11.30.Rd"],"model":"deepseek-v4-flash","headline":"In a mean-field quark-meson model, imposing finite-size constraints on momentum space shifts the critical endpoint of the QCD phase diagram significantly for system sizes below about 10 fm, and moves the baryon-fluctuation peak without…","keywords":["finite size effects","QCD phase diagram","critical endpoint","baryon fluctuations","quark-meson model","momentum space constraints","cumulant ratios","chiral symmetry breaking"],"falsifier":"Compute the baryon-number cumulant ratios, or the CEP location, for the same model (or for QCD) at $L=6$ fm using an independent finite-volume method, such as a direct spatial discretization or lattice QCD in a box; if the peak does not move to higher chemical potentials or if the shape changes substantially, the momentum-space constraint premise is wrong. Alternatively, compare data from small and large heavy-ion collision systems at the same beam energy and check whether the cumulant-ratio peak shifts as predicted.","tokens_in":9239,"feed_emoji":"⚛","tokens_out":10683,"duration_ms":87242,"temperature":0.7,"pith_summary":"Heavy-ion collisions create fireballs of finite size, while most effective-model calculations of the QCD phase diagram assume an infinite volume. This paper asks whether that mismatch matters, and answers yes: when finite size is imposed through momentum-space constraints, such as a low-momentum cutoff or mode discretization with periodic or antiperiodic boundary conditions, the critical endpoint shifts significantly for linear sizes below $L\\approx 10$ fm. The shift usually moves the endpoint to larger chemical potentials or to lower temperatures. The same constraints also move the peak in baryon-number cumulant ratios along the phase boundary, but leave its shape essentially unchanged. A sympathetic reader should care because these results imply that comparisons between effective-model phase diagrams and experimental fluctuation data must include finite-size effects.","feed_headline":"Finite size shifts the QCD critical endpoint below 10 fm","feed_subtitle":"The shift is large enough that comparisons of model phase diagrams to heavy-ion data need to account for fireball size.","key_machinery":"The central object is the set of momentum-space constraints that convert the infinite-volume momentum integrals of the mean-field quark-meson model into finite-volume equivalents. The low-momentum cutoff replaces integrals by $\\int d^3p/(2\\pi)^3\\,\\theta(p-\\lambda)$ with $\\lambda=\\pi/L$; discretization replaces them by sums over modes $p_i=2n_i\\pi/L$ (periodic) or $p_i=(2n_i+1)\\pi/L$ (antiperiodic), summed spherically with a kernel and renormalized via the UV-improved scheme. These constraints act on both the fermionic vacuum and the thermal fluctuations, and they determine the size dependence of the chiral condensate, the phase boundary, the location of the CEP, and the baryon-number susceptibilities.","core_discovery":"Working in the mean-field quark-meson model, the paper compares three ways to impose finite volume: a low-momentum cutoff with $\\lambda = \\pi/L$, momentum discretization with periodic boundary conditions ($p_i = 2n_i\\pi/L$), and antiperiodic boundary conditions ($p_i=(2n_i+1)\\pi/L$), with and without modifying the fermionic vacuum fluctuations. It finds that the choice of scenario and the vacuum treatment strongly affect the volume dependence of the critical endpoint: with a cutoff on the vacuum contribution the broken phase disappears at $L\\approx 2.5$ fm, while with discretization chiral symmetry breaking is enhanced for decreasing size. Despite these differences, in every scenario the CEP is significantly shifted for $L\\lesssim 10$ fm, in most cases toward larger chemical potentials or lower temperatures. Along the phase boundary, the kurtosis and skewness ratios ($C_4/C_2$ and $C_2/C_1$) show the peak indicating the CEP is displaced to higher chemical potentials for both the discretized and cutoff cases, but the shape of the signal is hardly modified, because the criticality remains intact at mean-field level when only momentum-space constraints are applied.","pith_inferences":["If the momentum-space constraints capture the physics of a real fireball, then experimental estimates of the CEP obtained from cumulant ratios should be corrected for system size before being confronted with infinite-volume model predictions.","The 'staircase' phase transitions caused by discrete modes crossing the Fermi surface suggest that at small $L$ new critical points unrelated to the infinite-volume CEP may dominate low-temperature, high-density fluctuations; this could be tested by looking for multiple peaks in susceptibility scans.","The strong sensitivity of the results to the vacuum-fluctuation treatment indicates that the choice among cutoff, periodic, and antiperiodic schemes is not a technical detail; a first-principles finite-volume benchmark, such as lattice QCD in a box, would be needed to decide which scenario is physical."],"forward_implications":["For fireball linear sizes below about 10 fm, finite-size effects must be included before an effective-model phase diagram can be compared with heavy-ion data.","The direction and magnitude of the CEP shift depend on the chosen momentum-space constraint and on whether vacuum fluctuations are modified, so those choices must be stated and justified.","The path of the CEP as $L$ decreases is largely fixed by its infinite-volume position; models with a lower-lying CEP can show an interchange of the leading critical point at small sizes.","Baryon-number cumulant ratios along the phase boundary can still be used to search for the CEP at finite size, because the peak shifts but keeps its shape."],"supporting_citations":[{"why":"Introduced the low-momentum cutoff and finite-volume implementation for quark-meson models, providing the scenario this paper compares with discretization.","marker":"[1]"},{"why":"Earlier implementation of multiple momentum-space constraints in the same mean-field model, the basis for the phase-diagram and fluctuation results presented here.","marker":"[30]"},{"why":"Ideal boson gas study that motivates the low-momentum cutoff as a proxy for a finite spatial extent.","marker":"[45]"},{"why":"Supplies the Polyakov-loop potential used to incorporate statistical confinement, needed for the correct kurtosis value in the broken phase.","marker":"[46]"},{"why":"Provides the $N_f=2+1$ parameter sets used for the phase diagram and the baryon fluctuation calculations.","marker":"[41]"},{"why":"Shows that cumulant ratios retain an implicit size dependence, the quantity this paper computes along the phase boundary.","marker":"[39]"}],"fun_headline_variants":["QCD critical point shifts for small systems under 10 fm","Finite-size method alters QCD critical endpoint below 10 fm","System size moves QCD critical point, method matters","Small QCD systems shift critical endpoint significantly","Critical endpoint shifts with system size and finite-size scheme"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results rest on the assumption that the finite-size physics of a heavy-ion fireball is faithfully captured by imposing momentum-space constraints, such as a low-momentum cutoff or mode discretization in a cubic box, on a mean-field quark-meson model.","fun_headline_variants_meta":{"raw":{"variants":["QCD critical point shifts for small systems under 10 fm","Finite-size method alters QCD critical endpoint below 10 fm","System size moves QCD critical point, method matters","Small QCD systems shift critical endpoint significantly","Critical endpoint shifts with system size and finite-size scheme"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000821,"raw_usage":{"total_tokens":3573,"prompt_tokens":902,"completion_tokens":2671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":2592}},"tokens_in":518,"tokens_out":2671,"duration_ms":18049,"temperature":1.0,"reasoning_tokens":2592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:48:20.439280+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the baryon-number cumulant ratios, or the CEP location, for the same model (or for QCD) at $L=6$ fm using an independent finite-volume method, such as a direct spatial discretization or lattice QCD in a box; if the peak does not move to higher chemical potentials or if the shape changes substantially, the momentum-space constraint premise is wrong. Alternatively, compare data from small and large heavy-ion collision systems at the same beam energy and check whether the cumulant-ratio peak shifts as predicted.","supporting_citations":[{"cited_title":"Finite volume corrections and low momentum cuts in the thermodynamics of quantum gases","cited_arxiv_id":"1611.03746","evidence_quote":"Ideal boson gas study that motivates the low-momentum cutoff as a proxy for a finite spatial extent."}],"review_version":1}