{"id":"c634e0a0-0b21-4355-80d7-6a28612bfb71","arxiv_id":"2506.04313","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A universal finite-energy upper bound on the integrated cross-section for identical scalars is derived; the conjectured saturating \"Froissart amplitude\" shows Regge trajectories, a rising cross-section, and annulus-like diffraction.","lead":"Physicists derive a bound on the total scattering cross-section at any energy using only analyticity, crossing symmetry, and unitarity, and they identify a special \"Froissart amplitude\" that may saturate it. The result connects abstract S-matrix consistency conditions to realistic features of high-energy proton collisions, such as a rising cross-section and a shrinking diffraction cone.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The plotted finite-energy exclusion bound rests on an uncertified extrapolation of max c2(t0); primal bootstrap solutions give lower bounds on the true maximum, so the universal bound in Fig. 1 is not rigorously established.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing point: the practical universal bound in Fig. 1 depends on the numerical extrapolation of max c2(t0) from finite-N primal solutions, with no dual gap or error certificate. My stress-test confirms this is the central concern for the paper's main quantitative claim. The analytical derivation of Eq. (12) is a valid formal inequality and would become a rigorous universal bound if a certified upper bound on max c2(t0) were available. The numerical bootstrap results for the Froissart amplitude and the Regge trajectories are interesting and internally plausible, but they are secondary to the headline bound and equally lack a rigorous error bar. Given the absence of a formal proof for the extrapolation, the appropriate verdict remains CONDITIONAL, as the reader stated. I am not moving the verdict because the analytical contribution is substantial and the numerical issue is explicitly acknowledged in the text (notably in the discussion of the 5% dual-bound agreement at one point). The proposed concrete test would settle whether the uncertified extrapolation actually changes the plotted bound; until then, the central claim should be treated as conditional on numerical convergence.","tokens_in":41415,"tokens_out":4047,"duration_ms":41738,"concrete_test":"Compute certified upper bounds on max c2(t0) using the dual SDP method of [46,80] on a fine grid of t0 in (0,4), especially near the values that minimize the bound in Eq. (12) for s in the range 10–10^4. Re-evaluate the exclusion bound in Fig. 1 with these certified values instead of the power-law extrapolation. If the shifted bound curve moves upward by less than ~10% across the plotted s-range, the extrapolation is validated and the factor-of-ten comparison to data survives; if the shift exceeds a factor of about two, the central quantitative claim fails and the paper should be revised to a conjectural bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (12) is a formal inequality: for any fixed c2k(t0), it bounds the integrated cross-section in terms of that coefficient. The paper's quantitative claim, however, requires plugging in the true maximum of c2(t0) over all gapped scalar theories. The maximum is not known analytically; the paper estimates it by power-law extrapolation of primal S-matrix bootstrap solutions at finite cutoff N (up to N=20, 814 parameters) in the ansatz (14). A primal feasible point provides a lower bound on the maximum of c2(t0), not an upper bound. Since the RHS of (12) is increasing in c2(t0), using a lower bound on max c2(t0) produces a candidate exclusion bound that is smaller than the true bound and can incorrectly exclude consistent theories whose c2(t0) lies above the extrapolated value. The paper cites a 5% agreement with a dual bound at the specific point t0=4/3 [80,81], but Fig. 1 is obtained by minimizing (12) over t0, and no certified upper bounds for the full curve max c2(t0) are provided. Without a dual certificate or an explicit error interval on the extrapolation, the 'optimal exclusion bound' in Fig. 1 is a numerical estimate, not a proven universal bound. The analytical skeleton of Section II is sound, but the headline comparison with pp and pbarp data is conditional on unverified convergence of the numerical maximization.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives an upper bound on the integrated 2->2 total cross-section for gapped identical scalar theories in d dimensions, Eq. (12), valid at finite s for any t0 in (0,4), even L, and integer k, once the low-energy dispersive coefficient c_{2k}(t0) is known. Setting c_{2k}(t0) to its maximum over consistent amplitudes yields a theory-independent exclusion bound. In d=4 the asymptotic limit reduces to the Froissart bound. The authors then compute max c_2(t0) numerically with the S-matrix bootstrap using a truncated multi-foliation ansatz (14) with up to N=20 and 814 parameters, extrapolate to N->infinity, and use the result to plot an 'optimal exclusion bound' compared with pp and pbarp data, claiming agreement within a factor of about ten for m about 1 GeV. They identify the extremal amplitude at the (min c0, max c2) cusp as the conjectured 'Froissart amplitude' and study its total cross-section, Regge trajectories, diffractive minima, and eikonal profile, proposing a 'white ring' annulus picture.","tokens_in":41750,"tokens_out":11397,"duration_ms":103526,"significance":"Eq. (12) is a clean and potentially important finite-energy generalization of the Froissart bound: it converts the asymptotic log^2 growth into a finite-s inequality controlled by a single positive low-energy coefficient, and it is derived transparently from unitarity, crossing, and dispersion relations. The numerical exploration is state-of-the-art and the paper is rich in phenomenology. If the extrapolated max c_2(t0) were replaced by certified dual bounds, the exclusion plot would be a rigorous universal statement; as it stands, the analytic derivation is the lasting core, while the 'factor of ten' and the detailed properties of the Froissart amplitude are numerical and conditional evidence. The paper should be credited for being explicit about many limitations (e.g., footnotes on numerical convergence), but those limitations are in tension with some abstract-level claims.","major_comments":[{"comment":"The input max c_2(t0) used in the exclusion bound is obtained by extrapolating finite-N primal bootstrap solutions. A primal feasible solution provides a lower bound on the true maximum, and since the right-hand side of Eq. (12) is increasing in c_2(t0), the red excluded region in Fig. 1 can lie below the true bound. The statement in Section III.A that the extrapolation agrees with a dual bound to 5% at t0=4/3 is a useful cross-check, but Fig. 1 is produced by minimizing Eq. (12) over t0, and no certified upper bounds for the full curve max c_2(t0) are given. The 'optimal exclusion bound' is therefore a numerical estimate, not a proven universal bound; the quantitative claim in the abstract and in Fig. 1 should be rephrased or supported by dual certificates.","section":"Section III.A, Figs. 1-2"},{"comment":"The Regge trajectories, the Pomeron-like trajectory B with intercept alpha0 about 1.1, the daughter trajectories, and the 'singular' trajectories Z_i are obtained by analytic continuation of the truncated N=20 amplitude. The paper itself notes in footnote 18 that the continuation near ell~1 may be affected by systematic numerical errors, and in Fig. 19 that trajectory B is the least stable. Despite this, the abstract states as a result a 'surprisingly rich spectrum of resonances aligning along Regge trajectories, including Pomeron-like and daughter trajectories'. These are numerical indications for a conjectured amplitude, not established properties; the language should be qualified accordingly.","section":"Section IV.B, Fig. 7, and abstract"},{"comment":"The 'white ring' picture is an interpretation based on fitting the phenomenological model (20) to the differential cross-section of the numerical amplitude. The eikonal representation (D1)-(D4) is an approximation valid at large s, and the fitted radii R1 and R2 are not derived from the bootstrap. Therefore the claim in the abstract that 'the eikonal representation reveals that the scattering is localized within an annular region' is model-dependent evidence, not a first-principles consequence. This should be made explicit in the main text, not only in the figure captions.","section":"Section IV.C, Figs. 8-9, Appendix D"},{"comment":"The existence of a spin-2 bound state at threshold is inferred from the growth of c_2(t0) as t0 approaches 4 and from the ell=2 phase-shift behavior at increasing N (Fig. 17), but the ansatz (14) contains no such singularity at s=4. The claim in Appendix G that 'we can confidently claim that the Froissart amplitude features a spin two threshold state' goes beyond what an extrapolation of finite-N data can establish. Since this divergence is also used to explain why the finite-s optimization avoids t0 approaching 4, the claim should be labelled as a numerical conjecture or supported by an explicit singular ansatz.","section":"Section III.A and Appendix G"}],"minor_comments":[{"comment":"The m about 1 GeV assumption underlying the comparison with pp data should be stated as an external physical input, not a consequence of the bootstrap; varying m shifts the bound by powers of m.","section":"Section I, Fig. 1"},{"comment":"In Fig. 2 and Fig. 3, the N-to-infinity extrapolation is shown without specifying the power-law fitting function, the fit range, or the systematic error; please provide these details or a reference.","section":"Section III.A, Figs. 2-3"},{"comment":"The caption of Fig. 7 says 'Blue dots denote the particles showing up at real values of the spin arranging into Regge trajectories'; this should be 'at non-integer values of the spin' to be consistent with the text and with Fig. 19.","section":"Section IV.B, Fig. 7"},{"comment":"The dual bound [80,81] is cited as 'work in progress' in [81]; if the dual calculation is available, include the relevant numbers and error bars in the text so the 5% cross-check can be assessed.","section":"Section III.A"},{"comment":"Equation (17) should define Disc_t T(s,t) and specify the integration contour; as written, the Froissart-Gribov formula requires more explanation for the non-expert reader.","section":"Section IV.B, Eq. (17)"},{"comment":"The footnote 'for some finite arbitrary even L' should say 'positive even L at least 2', since the derivation uses (L-1)^{d-2} and the partial-wave split at ell=L.","section":"Section II, Eq. (8)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong Letter and the analytic bound (12) is publishable. The main issue is the gap between the proven inequality and the unproven numerical extrapolation used in the headline plot. If the authors can provide dual upper bounds on max c_2(t0) over the relevant t0 range, or at least an honest error band and a softened abstract, the paper would be acceptable. The citation to [80,81] as 'work in progress' for the dual bound is a concern for a journal submission; a published or otherwise available certificate should be referenced."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the analytic result is real: Eq. (12) gives a strict finite-energy upper bound on the integrated total cross-section for gapped scalars in any d, using crossing, unitarity, and a single low-energy coefficient c2(t0). That removes Yndurain's spin-2 scattering-length assumption and recovers Froissart asymptotically. Second, the striking exclusion plot in Fig. 1 is not a proven universal bound, because it uses max c2(t0) estimated by power-law extrapolation of primal bootstrap solutions up to N=20. Primal solutions give lower bounds on the maximum, and the RHS of (12) grows with c2, so the plot is an estimate of where the true bound lies, not a certificate.\n\nWhat the paper does well: the derivation in Section II is clean, and the appendix on the d-dimensional Froissart bound is a useful counterpart to the existing literature. The numerical work is substantial: 814 parameters, careful foliation, improved positivity, and a reproducible setup with SDPB. The authors are honest about convergence issues in the Chew–Frautschi plot and the slow approach to the asymptotic limit. The \"white ring\" eikonal model is a nice heuristic, and the connection to Kupsch and Oehme is well placed. The paper also generalizes the bound to arbitrary test functions in Appendix I.\n\nThe soft spots: (1) the main quantitative claim—the factor-of-ten comparison with pp data—depends on the extrapolation of max c2(t0). The paper cites 5% agreement with a dual bound at t0=4/3, but no dual certificate for the full curve is provided. This should be fixed or clearly labeled as an estimate. (2) The Froissart amplitude's Regge spectrum, especially the singular trajectories and the Pomeron intercept, is extracted from a truncated ansatz and is not fully converged; the authors say so, but the conjectural status should be prominent in the abstract. (3) The spin-2 threshold bound state is inferred from indirect evidence rather than constructed; that is fine as a conjecture, but it is not a theorem.\n\nOverall: the analytic half of the paper is a solid contribution that deserves a serious referee. The numerical half is plausible and well executed but not yet rigorous. I would referee it, asking for dual bounds or softened claims about the optimal exclusion curve. The paper is worth discussing and citing.","headline":"The analytic bound is a real result; the exclusion plot is an extrapolation without a certificate, which is the one thing standing between this paper and a clean acceptance.","tokens_in":42264,"tokens_out":2822,"would_cite":true,"duration_ms":26750,"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":"A finite-energy bound on the total cross-section follows from unitarity and crossing symmetry alone, and lands within a factor of ten of proton-proton data.","keywords":["S-matrix bootstrap","Froissart bound","integrated total cross-section","Regge trajectories","Pomeron","eikonal approximation","scalar scattering","finite energy bounds"],"falsifier":"Compute a rigorous dual upper bound on $c_2(t_0)$ at $t_0 = 4/3$ (or any fixed $t_0$) using the method of [46]; if that dual bound falls strictly below the extrapolated value plotted in Fig. 2, the exclusion bound in Fig. 1 would have to be revised, since no consistent amplitude could reach the extrapolated $c_2$. Alternatively, construct or numerically find a bootstrap-allowed amplitude whose integrated cross-section at some finite $s$ exceeds the extrapolated bound of Fig. 3.","tokens_in":41219,"feed_emoji":"⚛️","tokens_out":8219,"duration_ms":69124,"temperature":0.7,"pith_summary":"This paper tries to show that the total scattering cross-section, integrated over energies up to a finite s, is bounded from above in any gapped scalar theory by a number that depends only on a single low-energy coefficient, with no assumption about the spin-2 scattering length. If the bound and its numerical evaluation are correct, the tightly constrained quantity sits within a factor of about ten of measured proton-proton and proton-antiproton cross-sections, making Froissart-type saturation quantitatively testable rather than asymptotic. The paper also identifies the amplitude that saturates the bound at high energies — the 'Froissart amplitude' — and claims it is a universal, strongly-coupled object with a rising cross-section, a shrinking diffractive cone, Regge trajectories including a Pomeron-like one, and an impact-parameter profile shaped like a growing ring rather than a disk. Such an object, if real, would give a first-principles handle on the soft, non-perturbative regime of QCD.","feed_headline":"Cross-section bound at finite energy within factor of ten of pp data","feed_subtitle":"A single low-energy coefficient sets the bound at all energies, not just asymptotically.","key_machinery":"The machinery is a two-step inequality. First, after splitting the partial-wave sum at a cutoff $L$, unitarity fixes the low-spin contribution by setting $\\mathrm{Re}\\, S_\\ell = -1$, while the high-spin tail is bounded by relating it, through Eq. (11), to the positive dispersive coefficient $c_{2k}(t_0) = \\frac{2^{1-k}}{\\pi}\\int_4^\\infty ds \\frac{\\mathrm{Im}\\,T(s,t_0)}{(s-2+t_0/2)^{2k+1}}$, defined by derivatives of the amplitude at the crossing-symmetric point. Optimizing over $L$ and $t_0$ yields Eq. (12), whose large-$s$ limit is the d-dimensional Froissart bound. The numerical part uses a multi-foliation ansatz (14) with unitarity imposed via positive semi-definite matrices and improved positivity constraints, solved with SDPB to obtain $\\max c_2(t_0)$ and the exclusion bound.","core_discovery":"The central discovery is an inequality, Eq. (12), that bounds the integrated total cross-section for identical scalar scattering at any finite energy s in any spacetime dimension d, using only unitarity, crossing symmetry, and polynomial boundedness of the amplitude. The bound is expressed in terms of the dispersive coefficient $c_{2k}(t_0)$, which is positive by a dispersion relation and can be bounded from above numerically by the S-matrix bootstrap. Removing the finiteness assumption on the D-wave scattering length made by Yndurain, the bound is universal for gapped scalars; in the asymptotic limit it reduces to the d-dimensional Froissart bound with the correct $\\log^{d-2} s$ growth. Numerically, the authors maximize the integrated cross-section directly and conjecture that the extremal amplitude coincides in the high-energy limit with the universal amplitude that maximizes all low-energy coefficients $c_{2k}(t_0)$; this 'Froissart amplitude' is then exhibited in detail, with its Regge spectrum, diffractive minima, and eikonal ring profile.","pith_inferences":["I infer that if the white ring picture is generic, the asymptotic elastic-to-total ratio need not approach the black disk value $1/2$; inelastic scattering would be increasingly peripheral, offering a bootstrap-derived analogue of the 'hollowness' effect discussed phenomenologically.","The conjectured identity between the Froissart amplitude and the $\\min c_0/\\max c_2$ cusp is testable within the bootstrap: any other objective that drives amplitudes to the same cusp (e.g., maximizing higher $c_{2k}$) should produce the same Regge spectrum, a check the authors have not performed.","I infer that extending the same bound to fermionic (proton-like) amplitudes would likely modify the intercept structure and potentially produce an Odderon trajectory, since Bose symmetry is what currently removes odd spins.","The bound's dependence on a single low-energy coefficient suggests a concrete test: measuring the Wilson coefficient $c_2(t_0)$ in a candidate theory fixes the maximal integrated cross-section, so a computed cross-section above the bound would falsify the S-matrix assumptions."],"forward_implications":["Any consistent gapped scalar theory in $d$ dimensions has an integrated total cross-section bounded at finite $s$ by Eq. (12); the bound is strict and requires no assumption on the spin-2 scattering length.","In four dimensions, with $m \\approx 1\\,\\mathrm{GeV}$, the optimal exclusion bound lies within a factor of about ten of measured $pp$ and $\\bar p p$ integrated cross-sections, making the bound phenomenologically relevant rather than purely asymptotic.","The amplitude that maximizes the integrated cross-section at high energies is conjectured to be the universal Froissart amplitude; it exhibits a rising total cross-section, a shrinking diffractive cone with multiple minima, and Regge trajectories, one with intercept near 1.15 resembling the phenomenological Pomeron.","The eikonal profile of the Froissart amplitude is a slowly expanding annulus, so Froissart growth is realised by a growing ring rather than a growing disk; a simple 'white ring' model reproduces the diffractive pattern and predicts a Regge cut with branch points at $\\ell = 1 \\pm r_0 \\sqrt{t}$.","The Froissart amplitude carries a spin-2 bound state at threshold, which diverges in the $t_0 \\to 4$ limit, explaining why the finite-energy optimal $t_0$ cannot be taken at threshold."],"supporting_citations":[{"why":"The original Froissart bound that this paper upgrades to a finite-energy, dimension-general statement.","marker":"[5]"},{"why":"Yndurain's earlier integrated cross-section bound, whose spin-2 scattering length assumption is removed here.","marker":"[74]"},{"why":"Provides the S-matrix bootstrap framework and the ansatz (14) used in the numerics.","marker":"[75]"},{"why":"Supplies the improved positivity constraints (16) that stabilize the numerical bootstrap at finite spin cutoff.","marker":"[48]"},{"why":"Phenomenological fits to pp and pbarp total cross-sections used for the data comparison in Fig. 1.","marker":"[76]"},{"why":"Establishes the positive dispersive representation of the coefficient c_{2k}(t0) used in the bound.","marker":"[77]"},{"why":"The SDPB solver used to perform the semidefinite bootstrap optimizations.","marker":"[79]"},{"why":"Provides a dual bound on c2(4/3) against which the extrapolated max c2 is compared.","marker":"[80]"},{"why":"Method for deriving dual bounds on c2 and for identifying the universal amplitude at the strong-coupling corner.","marker":"[46]"}],"fun_headline_variants":["S-matrix bootstrap bounds cross-section at finite energies","One low-energy coefficient sets cross-section bound at all energies","Froissart amplitude from bootstrap shows diffractive rings","Bootstrap amplitude matches pp data within factor of ten"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The plotted 'optimal exclusion bound' in Fig. 1 uses a value of $\\max c_2(t_0)$ obtained by extrapolating truncated numerical bootstrap solutions (up to $N=20$, with 814 parameters) to $N\\to\\infty$ with a power-law fit, and that extrapolation has no rigorous error certificate; if the true infinite-$N$ maximum is lower, the practical bound weakens, though equation (12) remains a valid bound for whatever $c_2$ one inserts.","fun_headline_variants_meta":{"raw":{"variants":["S-matrix bootstrap bounds cross-section at finite energies","One low-energy coefficient sets cross-section bound at all energies","Froissart amplitude from bootstrap shows diffractive rings","Bootstrap amplitude matches pp data within factor of ten"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000683,"raw_usage":{"total_tokens":3107,"prompt_tokens":961,"completion_tokens":2146,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":2081}},"tokens_in":577,"tokens_out":2146,"duration_ms":18377,"temperature":1.0,"reasoning_tokens":2081,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:44:58.497527+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a rigorous dual upper bound on $c_2(t_0)$ at $t_0 = 4/3$ (or any fixed $t_0$) using the method of [46]; if that dual bound falls strictly below the extrapolated value plotted in Fig. 2, the exclusion bound in Fig. 1 would have to be revised, since no consistent amplitude could reach the extrapolated $c_2$. Alternatively, construct or numerically find a bootstrap-allowed amplitude whose integrated cross-section at some finite $s$ exceeds the extrapolated bound of Fig. 3.","supporting_citations":[{"cited_title":"QCD Worldsheet Axion from the Bootstrap","cited_arxiv_id":"2310.20698","evidence_quote":"Provides the S-matrix bootstrap framework and the ansatz (14) used in the numerics."}],"review_version":1}