{"id":"0f714693-873c-44df-8d85-50fadd2aabe0","arxiv_id":"2502.01911","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A new Levy alpha-stable version of the ReBB elastic scattering model is derived, recovering the original for alpha=2, with no fits yet.","lead":"A generalized model for elastic proton-proton and proton-antiproton scattering is proposed, replacing Gaussian ingredient shapes with Levy alpha-stable ones. The paper gives the equations but does not yet fit data, and argues the new model may reconcile low-|t| and high-|t| measurements.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lévy-stable generalization violates unitarity at zero impact for α_L<2 because σ_ab^in(0)=A_ab·2Γ(2/α_L)/α_L exceeds 1 unless A_ab is constrained; with A_qq=1 the intended regime α_L<2 is excluded.","rationale":"The reader identified the same load-bearing assumption: the unaddressed unitarity bound on σ_ab^in for α_L<2. My independent analysis confirms it quantitatively from the paper's own Eq. (2), including the specific factor 2Γ(2/α_L)/α_L. This is not merely a parameter-region nuisance; it blocks the claimed regime α_L<2 when A_qq retains its ReBB value of 1. The paper is transparent about the absence of fits and positions itself as a model proposal, so the lack of data comparison is not the central issue. The check is analytic and inexpensive: re-derive the zero-impact value and impose the bound. The reader's CONDITIONAL verdict is appropriate: the construction is clear and the generalization is novel, but the model must be amended with explicit constraints on A_ab or a proof that the unitarity bound is satisfied. No change to the reader's verdict is needed; the condition is confirmed rather than shifted.","tokens_in":4914,"tokens_out":7814,"duration_ms":75930,"concrete_test":"Derive the peak condition from Eq. (2): for fixed α_L, require A_ab·2Γ(2/α_L)/α_L ≤ 1 for each pair (a,b)∈{q,d}². Numerically evaluate this for α_L=1.959 with A_qq=1; the left side is ≈1.009>1, so the ReBB value violates unitarity. Then determine, for α_L∈[1.5,2], the maximal allowed A_qq; if the resulting constraint is incompatible with the intended interpretation of A_qq as a probability (e.g. A_qq<0.94 for α_L=1.5), the model must be amended by either letting A_qq float with an explicit bound or by proving the Glauber average in Eq. (4) still keeps σ̅_in≤1 despite σ_ab^in(0)>1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing assumption is that the constituent-level inelastic probability σ_ab^in(s) from Eq. (2) remains in [0,1] for α_L<2, so that the Glauber product in Eq. (1) and the unitarized amplitude in Eq. (5) are well defined. For the two-dimensional symmetric stable density used here, L(0|α,R)=Γ(2/α)/(2π α R^2); applying the convolution identity in Eq. (2) gives σ_ab^in(0)=A_ab f(α_L) with f(α)=2Γ(2/α)/α. Since f(2)=1 but f(α)>1 for every α<2 (e.g. f(1.959)≈1.009, f(1.8)≈1.07), a constituent probability exceeds 1 at zero impact if A_ab is fixed at the ReBB value A_qq=1. The paper neither checks this bound nor constrains A_ab, although α_L is introduced precisely to move below 2. Because L is continuous and has its maximum at 0, this violation occurs on a set of positive measure, so it is not a negligible point-wise artifact. If σ_ab^in>1, the product (1-σ_ab^in) can become negative, and the proton-level σ̅_in can exceed 1, making √(1-σ̅_in) in Eq. (5) imaginary and the amplitude non-unitary. This is an internal correctness risk, not a disagreement with external consensus: with A_qq=1 the model cannot have α_L<2 at all, so the central motivation of the paper is unsupported unless the parameter space is restricted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This short proceedings paper introduces a Lévy α-stable generalization of the Bialas–Bzdak (ReBB) model of elastic proton-proton and proton-antiproton scattering. The constituent-constituent inelastic scattering probabilities of Eq. (2) and the quark-diquark distribution of Eq. (3) are replaced by symmetric Lévy α-stable distributions, with the same stability index α_L for all ingredients. The paper notes that α_L = 2 recovers the Gaussian ReBB model, and it argues that, based on earlier low-|t| analyses, α_L < 2 should allow a simultaneous description of low- and high-|t| data. No data fitting is performed; the paper is a model proposal together with motivation from the authors' previous work.","tokens_in":5302,"tokens_out":5168,"duration_ms":49978,"significance":"If the LBB model indeed describes both the low-|t| non-exponential behavior and the high-|t| dip region of elastic pp and p̄p scattering with a single value of α_L < 2, it would be a useful tool for studying the ATLAS–TOTEM tension and the Odderon contribution. The mathematical generalization is cleanly specified: the convolution identity in Eq. (2) is valid for symmetric stable distributions, the normalization of Eq. (3) is correct, and the α_L = 2 limit reproduces the Gaussian ReBB forms. The paper is honest about being a proposal and explicitly says that applying the model to data is the next step. Its significance is prospective rather than demonstrated; no new experimental insight is obtained yet. The main weakness is the absence of any check that the constituent-level probabilities remain physical for α_L < 2.","major_comments":[{"comment":"The paper does not enforce the probabilistic bound 0 ≤ σ_ab^in ≤ 1. For the symmetric stable density normalized as L(0|α,R) = Γ(2/α)/(2π α R^2), the zero-impact value is σ_ab^in(0) = A_ab f(α_L) with f(α) = 2Γ(2/α)/α. Since f(2)=1 and f(α)>1 for every α<2, using the ReBB value A_qq=1 (as in the fits displayed in Figs. 1 and 2) gives σ_qq^in > 1 at b=0 for the very regime α_L<2 that motivates the paper. Because the stable density is continuous and maximal at 0, the violation occurs on a set of positive measure, so (1−σ_ab^in) in Eq. (1) can become negative and the square root in Eq. (5) imaginary. The manuscript should state and impose the constraint A_ab ≤ α_L/[2Γ(2/α_L)] and discuss whether the expected α_L ≈ 1.959 is compatible with the ReBB calibration A_qq=1.","section":"Section 3, Eq. (2)"},{"comment":"The Summary asserts that the LBB model 'is expected to describe simultaneously the low-|t| and high-|t| domains of elastic pp and p̄p dσ/dt' with α_L < 2. This is presented as a central outcome of the paper, but no data comparison, no fit, and no model calculation of a differential cross section is performed anywhere in the manuscript. The expectation is a hypothesis motivated by the ReBB results and by Ref. [8], not a demonstrated property of the LBB model. Please rephrase the claim as a program to be carried out in future work, and specify the intended kinematic domains and the fitting procedure already at the proposal stage.","section":"Section 4 (Summary)"}],"minor_comments":[{"comment":"The text says '√s is the squared center of mass energy'; this should read '√s is the center-of-mass energy'.","section":"Section 2, second paragraph"},{"comment":"The Fourier convention for L(⃗x|α_L,R_L) should be stated explicitly, e.g., with the measure d²q/(2π)², so that the normalization check leading to L(0|α,R)=Γ(2/α)/(2π α R^2) is transparent.","section":"Section 3, Eq. (2)"},{"comment":"The captions do not identify which symbols correspond to the TOTEM and ATLAS data and which curves are the ReBB fit; please add explicit legends or state this in the captions.","section":"Figures 1 and 2"},{"comment":"The sentence 'The power of a simple Lévy α-stable model for elastic scattering was demonstrated in Ref. [8]' should explicitly repeat that the demonstration concerns low-|t| data only, as done earlier in the same paragraph, to avoid giving the impression that high-|t| data were already described by that model.","section":"Section 3, final paragraph"},{"comment":"In the Introduction, 'studies published in 2021 and 2022 [3, 4]' is slightly misleading because Ref. [4] is dated 2021; please correct the year description or the citation grouping.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is a brief proceedings-style contribution whose central derivation is mathematically sound but whose physical regime is not fully specified. The missing unitarity bound on σ_ab^in is the main correctness issue; it is fixable by constraining A_ab, but it is load-bearing because the paper's motivation is precisely α_L < 2. Once the constraint is added and the Summary is rephrased as a future program, the paper would be suitable for publication. The scope of the journal should also be considered, since no data analysis is presented."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a clean, clearly written model-generalization paper that does what it says, but the new ingredient (α_L<2) conflicts with the probabilistic interpretation of the constituent-level cross section unless the normalization A_ab is constrained. The paper does not check this, and with A_qq=1 (the ReBB value) the intended regime α_L<2 is actually excluded. That is the soft spot that needs fixing before the model is used.\n\nWhat's new: replacing Gaussian shapes with symmetric Lévy α-stable distributions in both the constituent-constituent inelastic overlap and the quark-diquark distribution. The convolution identity is correctly applied, the α=2 limit recovers ReBB, and the paper is honest that no fits are performed yet. The motivation is sensible—Levy stable distributions are natural when heavy-tailed parton distributions show up, and their convolution property fits the Glauber framework well.\n\nSoft spots: (1) The stress-test concern is real and, as far as I can tell, correct. For α<2, the 2D symmetric stable density at the origin is Γ(2/α)/(2π α R^2), so σ_ab^in(0)=A_ab·2Γ(2/α)/α. For α=1.959 this is 1.009 A_ab; at α=1.8 it is about 1.07 A_ab. Since previous ReBB fits fix A_qq=1.000, the LBB model as parameterized cannot have α_L<2 without violating σ_ab^in≤1 on a set of positive measure. The paper neither states constraints on A_ab nor proves that the Glauber product and Eq. (5) remain real. This is a genuine internal consistency issue, not a disagreement with external consensus. (2) The 'expected to describe' claim is a hope, not a result; the paper contains no fits, which the authors acknowledge. That is not a flaw per se, but it means the paper is a proposal, not a demonstrated description.\n\nWho this is for: the hadronic scattering community, especially people working on the ATLAS-TOTEM low-|t| discrepancy and Odderon extraction. The construction is worth a serious referee because the math is transparent and the problem is fixable. A revision should explicitly restrict the parameter space (e.g., choose A_ab small enough to keep σ_ab^in≤1 for the intended α_L range) or prove that the unitarized amplitude remains well defined despite pointwise violations.\n\nRecommendation: send this to peer review. The flaw is real but not fatal to the idea; it is a missing constraint that a referee can ask to be added.","headline":"Clean model extension, but the Lévy parameter α_L<2 violates the unitarity bound on constituent probabilities unless A_ab is re-constrained.","tokens_in":5839,"tokens_out":3886,"would_cite":false,"duration_ms":36465,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The Lév y α-stable generalization of the Bialas-Bzdak (LBB) model replaces the Gaussian distributions in the ReBB model with Lévy α-stable shapes, recovering ReBB at α_L = 2 and aiming to describe low- and high-|t| elastic pp and p\\bar{p}…","keywords":["Lévy alpha-stable distribution","ReBB model","Bialas-Bzdak model","elastic proton-proton scattering","elastic proton-antiproton scattering","Glauber multiple scattering","TOTEM low-|t| non-exponential behavior","differential cross section"],"falsifier":"Fit the LBB model to the combined TOTEM low-|t| and high-|t| elastic pp data at $\\sqrt{s} = 8$ TeV: if the best-fit Lév y index $\\alpha_L$ is statistically compatible with 2, or if the fit does not reach CL $\\geq 0.1\\%$, or if any extracted $A_{ab}$ makes the constituent-constituent inelastic probability $\\sigma_{ab}^{in}(0)$ exceed 1, the paper's central expectation is falsified.","tokens_in":4705,"feed_emoji":"⚛️","tokens_out":6345,"duration_ms":56702,"temperature":0.7,"pith_summary":"The paper generalizes the Real extended Bialas-Bzdak (ReBB) model of elastic proton-proton and proton-antiproton scattering by replacing Gaussian shapes with Lévy α-stable distributions. Two ingredients are changed: the inelastic scattering probabilities of two constituents and the quark-diquark distribution inside the proton. When the Lévy index α_L equals 2, the Gaussian-based ReBB model is recovered, so the new LBB model contains the old one as a special case. The expected payoff is a simultaneous description of the low-|t| and high-|t| differential cross sections with α_L < 2, which the Gaussian ReBB model cannot achieve for TOTEM data at √s = 8 TeV.","feed_headline":"Gaussian shapes swapped for Lévy tails in proton scattering model","feed_subtitle":"One stability index α_L < 2 may reconcile TOTEM low-|t| and high-|t| data the ReBB model cannot fit together.","key_machinery":"The central object is the Lévy α-stable distribution $L(\\vec{x}|\\alpha_L, R_L) = \\frac{1}{(2\\pi)^2} \\int d^2\\vec{q}\\, e^{i\\vec{q}\\cdot\\vec{x}} e^{-|q^2 R_L^2|^{\\alpha_L/2}}$, which replaces the Gaussian both in the constituent-constituent inelastic probability (Eq. 2) and in the quark-diquark distribution (Eq. 3). Its defining convolution stability keeps the Glauber expansion algebraically closed: the inelastic probability for two constituents is again a Lévy α-stable function with the combined scale $S_{ab}$. The elastic amplitude is then obtained by averaging over constituent positions and applying the unitary real-part formula $\\tilde{T}_{el}(s,b) = i\\left(1 - e^{i\\alpha_R \\tilde{\\sigma}_{in}(s,b)}\\sqrt{1 - \\tilde{\\sigma}_{in}(s,b)}\\right)$, followed by a Fourier transform to momentum space.","core_discovery":"The central claim is that the Lévy α-stable generalization of the Bialas-Bzdak model is made by changing both (i) the inelastic scattering probabilities of two constituents and (ii) the quark-diquark distribution inside the proton from Gaussian shapes to Lévy α-stable shapes. The constituent-constituent inelastic probability is defined as a convolution of two Lévy α-stable distributions, which is again Lévy α-stable with scale parameter $S_{ab} = (R_a^{\\alpha_L} + R_b^{\\alpha_L})^{1/\\alpha_L}$, and the proton's internal distribution is a Lévy α-stable separation distribution between the quark and the diquark. The model reduces to the ReBB model when $\\alpha_L = 2$, and the paper argues that the new free parameter $\\alpha_L$ is expected to come out below 2, motivated by the strong non-exponential low-|t| behavior seen by TOTEM and by the good performance of a simpler Lévy α-stable model with $\\alpha \\approx 1.959$.","pith_inferences":["The same Lév y α-stable replacement of Gaussian distributions could be applied to other hadronic scattering models built on Glauber expansions, potentially inheriting the same unification of low- and high-|t| behavior.","A physically meaningful $\\alpha_L < 2$ may indicate that the proton's parton cloud has power-law tails, connecting the model's parameters to generalized central-limit-theorem arguments rather than being purely phenomenological.","A necessary check not performed in this paper is that $\\sigma_{ab}^{in}(\\vec{s})$ stays within [0,1] for all allowed parameters with $\\alpha_L < 2$; for Lév y distributions the peak at $\\vec{s}=0$ grows with $\\Gamma(2/\\alpha_L)/\\alpha_L$, so the normalization $A_{ab}$ must be small enough to preserve unitarity.","The model can be tested at other LHC energies (e.g., 2.76 and 13 TeV) to see whether a single $\\alpha_L$ describes the energy evolution or whether the Lév y index itself runs with $\\sqrt{s}$."],"forward_implications":["The LBB model should describe simultaneously the low-|t| and high-|t| domains of elastic pp and p\\bar{p} differential cross sections with a Lévy index $\\alpha_L < 2$, resolving the statistical failure of the ReBB model at √s = 8 TeV.","Because $\\alpha_L < 2$ produces heavier tails in the impact-parameter distribution, the model can generate the strong non-exponential low-|t| behavior observed by TOTEM, which the Gaussian ReBB model cannot reproduce.","After fits reach statistical acceptance, the model can be used to study the discrepancy between ATLAS and TOTEM total cross-section measurements.","With Coulomb-nuclear interference effects included, the model can extract the Odderon contribution to the parameter $\\rho_0$ at √s = 13 TeV.","If fits yield $\\alpha_L$ near the value around 1.959 seen in the simpler model, that would support the Lévy α-stable framework across SPS, Tevatron, and LHC energies."],"supporting_citations":[{"why":"Supplies the original Bialas-Bzdak model with Gaussian constituent distributions that the LBB model generalizes.","marker":"[1]"},{"why":"Supplies the Glauber multiple-diffractive-scattering expansion, Eq. (1), that builds the two-proton inelastic probability from constituent-constituent collisions.","marker":"[2]"},{"why":"Introduces the ReBB model, the real-extended version with a unitary real part, whose Gaussian shapes are replaced in the LBB model.","marker":"[3]"},{"why":"Establishes that the ReBB model describes pp and p\\bar{p} elastic data in a statistically acceptable manner, providing the baseline that fails at 8 TeV.","marker":"[4]"},{"why":"Documents the TOTEM observation of strong non-exponential low-|t| behavior with significance exceeding 7σ, the key motivation for leaving Gaussian shapes.","marker":"[6]"},{"why":"Provides the Lév y α-stable forms of the constituent inelastic probability and the quark-diquark distribution (Eqs. 2-3) that the LBB model builds on.","marker":"[7]"},{"why":"Shows that a simpler Lév y α-stable model describes low-|t| elastic data with $\\alpha \\approx 1.959$, motivating the expectation that $\\alpha_L < 2$.","marker":"[8]"}],"fun_headline_variants":["Proton scattering model trades Gaussians for Lévy α-stable","New parameter α_L may reconcile proton data ReBB can't","α_L<2: Lévy twist may unify proton data","ReBB goes Lévy: α_L<2 for proton data","Lévy α-stable upgrade to ReBB model for proton scattering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the inelastic constituent-constituent scattering probability defined by the Lév y α-stable convolution remains between 0 and 1 for all parameter values with $\\alpha_L < 2$, so that the Glauber expansion and the unitarized amplitude describe a physical probability.","fun_headline_variants_meta":{"raw":{"variants":["Proton scattering model trades Gaussians for Lévy α-stable","New parameter α_L may reconcile proton data ReBB can't","α_L<2: Lévy twist may unify proton data","ReBB goes Lévy: α_L<2 for proton data","Lévy α-stable upgrade to ReBB model for proton scattering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001483,"raw_usage":{"total_tokens":5886,"prompt_tokens":803,"completion_tokens":5083,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":419,"completion_tokens_details":{"reasoning_tokens":4992}},"tokens_in":419,"tokens_out":5083,"duration_ms":35670,"temperature":1.0,"reasoning_tokens":4992,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:00:25.033735+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fit the LBB model to the combined TOTEM low-|t| and high-|t| elastic pp data at $\\sqrt{s} = 8$ TeV: if the best-fit Lév y index $\\alpha_L$ is statistically compatible with 2, or if the fit does not reach CL $\\geq 0.1\\%$, or if any extracted $A_{ab}$ makes the constituent-constituent inelastic probability $\\sigma_{ab}^{in}(0)$ exceed 1, the paper's central expectation is falsified.","supporting_citations":[{"cited_title":"Bialas and A","cited_arxiv_id":null,"evidence_quote":"Supplies the original Bialas-Bzdak model with Gaussian constituent distributions that the LBB model generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Glauber multiple-diffractive-scattering expansion, Eq. (1), that builds the two-proton inelastic probability from constituent-constituent collisions."},{"cited_title":"Nemes, T","cited_arxiv_id":null,"evidence_quote":"Introduces the ReBB model, the real-extended version with a unitary real part, whose Gaussian shapes are replaced in the LBB model."},{"cited_title":"Cs¨ org˝ o and I","cited_arxiv_id":null,"evidence_quote":"Establishes that the ReBB model describes pp and p\\bar{p} elastic data in a statistically acceptable manner, providing the baseline that fails at 8 TeV."},{"cited_title":"Antchev et al","cited_arxiv_id":null,"evidence_quote":"Documents the TOTEM observation of strong non-exponential low-|t| behavior with significance exceeding 7σ, the key motivation for leaving Gaussian shapes."},{"cited_title":"Cs¨ org˝ o, S","cited_arxiv_id":null,"evidence_quote":"Provides the Lév y α-stable forms of the constituent inelastic probability and the quark-diquark distribution (Eqs. 2-3) that the LBB model builds on."},{"cited_title":"Cs¨ org˝ o, S","cited_arxiv_id":null,"evidence_quote":"Shows that a simpler Lév y α-stable model describes low-|t| elastic data with $\\alpha \\approx 1.959$, motivating the expectation that $\\alpha_L < 2$."}],"review_version":1}