{"id":"cf748f85-afa4-4fd6-8ca0-57c40a92d809","arxiv_id":"2504.18841","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Geometric scaling still holds for the dip-bump region of elastic pp scattering at the LHC, and crossing symmetry yields a one-parameter formula for the rho parameter and the bump-to-dip ratio.","lead":"This paper argues that geometric scaling, a 1970s scaling law for elastic proton-proton scattering, still holds at LHC energies in the region of the dip and bump of the differential cross section. It uses crossing symmetry to derive formulas for the rho parameter and the bump-to-dip ratio, which mostly match the data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The odderon-free crossing derivation produces a rho prediction that is inconsistent with the TOTEM 13 TeV measurement; because Eq. (26) uses this same rho, the Rbd prediction is not as robust as it appears.","rationale":"The reader's weakest_assumption identifies the odderon-free assumption and the asymptotic expansion, which is the same general area as my concern. However, I sharpen the issue: the odderon-free assumption is not merely a completeness caveat, because Eq. (19), derived from the same crossing relation, visibly overshoots the low TOTEM 13-TeV rho point, and the Rbd prediction in Eq. (26) depends explicitly on this same rho. Thus the central claim, that formulas (19) and (26) reproduce the dip-bump data, cannot be fully separated from a failed quantitative prediction of the same framework. The paper does acknowledge the issue and gives a fair discussion, which supports a conditional rather than a reject verdict. My agreement is partial because the reader framed the issue as a missing extension, whereas I view it as an internal tension between a successful Rbd prediction and a failing rho prediction from the same model, which deserves a direct quantitative resolution. Both the PDG and DL parametrizations overshoot the TOTEM 13-TeV rho measurement, so the inconsistency is not an artifact of one parametrization. The same concern applies to the dip-bump ratio because Rbd = c0*(1+rho^2)/rho^2 and c0 is fitted, so an rho that is too high lowers the predicted Rbd. For rho ~ 0.13, a 30 percent change in rho changes Rbd by roughly 15 percent, comparable to the spread of the plotted data points, so the effect is not negligible. I therefore retain the CONDITIONAL verdict and ask for either an odderon-inclusive computation of Rbd or an explicit error analysis showing that the TOTEM rho discrepancy has negligible impact on the Rbd comparison.","tokens_in":8682,"tokens_out":2907,"duration_ms":21673,"concrete_test":"Refit the Rbd predictions including an explicit C-odd contribution to the amplitude, parametrized minimally as O(s,|t|) = i*s*rho_O(y)*Im T(s,|t|) with rho_O(y) tuned to the TOTEM 13-TeV rho measurement, and recompute the ratio Rbd from the position-dependent real and imaginary parts near the dip and bump. Alternatively, recompute Eq. (19) with the ATLAS 13-TeV rho measurement instead of TOTEM and check whether the implied change in rho shifts the Rbd curve by more than the quoted uncertainties on the plotted data.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central quantitative results, Eqs. (19) and (26), follow from identifying Re T via crossing of the GS amplitude (14). Two load-bearing assumptions are involved: (i) purely C-even amplitude with no odderon; (ii) first-order expansion in y-i*pi/2 expressed in Eqs. (16)-(17). The paper itself acknowledges (i) in Sec. 3: the low TOTEM rho points at 13 TeV, which the model visibly overshoots, have been interpreted as odderon evidence, and the author states the analysis should be extended to C-odd amplitudes. Because the same crossing machinery determines both rho and the dip-bump ratio Rbd, the disagreement between predicted rho, Eq. (19), and the 13-TeV TOTEM rho measurement is directly relevant to the Rbd prediction: the predicted Rbd = c0*(1+rho^2)/rho^2 at LHC energies depends on the same rho that Eq. (19) overshoots. If the true rho is lower due to an odderon, the Rbd curve in Fig. 4 would be shifted relative to the data. Assumption (ii) is also explicitly asymptotic; the paper notes the rho formulas overshoot ISR points. While Rbd is only weakly sensitive to rho at LHC values, this does not remove the concern: the derivation makes a definite prediction for rho that fails at the highest energy point where the odderon question is most acute, and the dip-bump ratio is computed through the same amplitude. The paper presents the TOTEM rho discrepancy as context rather than as a test that impacts Rbd, but the logical connection is direct and unresolved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript argues that geometric scaling (GS), with the scaling variable tau proportional to sigma_tot(s)|t|, persists at LHC energies in the dip-bump region of elastic pp differential cross sections, despite its established violation for integrated cross sections. The evidence is the constancy of the bump-to-dip position ratio Tbd = 1.355 +/- 0.011 from ISR to LHC energies. The author then uses crossing symmetry and a first-order Taylor expansion in the imaginary rapidity shift to separate real and imaginary parts of the amplitude, deriving a formula for the rho parameter and a two-parameter form for the bump-to-dip cross-section ratio Rbd. These formulas are compared with data using two total cross-section parametrizations, and the paper also discusses the failure of GS to describe the energy dependence of the total elastic cross section, attributing it to GS violation at small |t|. The paper is explicitly qualitative in nature and acknowledges the odderon caveat and the asymptotic character of the expansion.","tokens_in":9027,"tokens_out":4485,"duration_ms":48385,"significance":"If the empirical scaling claim holds, the paper documents a striking regularity across five decades of energy and provides a simple analytical framework connecting this scaling to the real part of the amplitude. The constant Tbd observation is a robust, model-independent result that deserves attention. The derivation of rho from the energy derivative of R^2 is elegant and connects to older dispersion-relation results, but its practical predictive power is limited by the need for external total cross-section parametrizations, by the fitted constant c0, and by the explicit neglect of C-odd contributions. The paper is honest about these limitations, which is a strength, but the presentation sometimes overstates the status of Eqs. (19) and (26) as predictions.","major_comments":[{"comment":"The constant c0 in Eq. (27) is not derived from a known profile function Phi; the values c0 = 0.012-0.013 used in Fig. 4 are chosen to match the observed Rbd data. Therefore the agreement in Fig. 4 is a two-parameter fit (c0 plus the chosen total cross-section parametrization), not the 'one parameter prediction' stated in the text. The abstract and Section 3 should state this explicitly, since the current wording implies more predictive power than the model actually has.","section":"Sec. 3, Eq. (26) and Fig. 4"},{"comment":"The derivation of rho uses the crossing relation (13), which assumes a purely C-even amplitude with no odderon contribution. The paper notes that the low TOTEM rho points at 13 TeV may signal an odderon. Because Eq. (26) for Rbd depends on the same rho, the discrepancy between Eq. (19) and the TOTEM measurement is directly relevant to the Rbd prediction: a lower rho would shift the Rbd curves upward relative to the data. The paper should either include a C-odd term in the amplitude or quantitatively estimate how the rho uncertainty propagates into Rbd. As written, the two results are not independent, and the failure of Eq. (19) at the highest energy point weakens the claim that Eq. (26) is supported by the data.","section":"Sec. 3, Eqs. (19) and (26)"},{"comment":"The first-order Taylor expansion in the imaginary rapidity shift, Eqs. (16)-(17), is asymptotic, and the paper itself notes that the ISR rho data are overshot. While this is acknowledged in Section 3, the derivation in Section 2 should state at the outset that the resulting formulas for rho and Rbd are only expected to hold at sufficiently high energy where the expansion converges. The current comparison against the full ISR-to-LHC range in Figs. 3 and 4 gives the impression of a quantitative reproduction that is not supported by the accuracy of the expansion.","section":"Sec. 2, Eqs. (16)-(17) and Fig. 3"}],"minor_comments":[{"comment":"In Eq. (23), the left-hand side should refer to tau_bump rather than tau_dip, and the notation R^2_1(y) appears without definition; it should presumably be R^2(y).","section":"Sec. 2, Eq. (23)"},{"comment":"There are several typographical errors, including 'gown' for 'down' in Section 4, 'tis' for 'this', and 'limitted' for 'limited'. These should be corrected.","section":"Throughout"},{"comment":"The caption of Fig. 4 does not identify which curve corresponds to the PDG parametrization (20) and which to the DL parametrization (21); the legend should be added.","section":"Fig. 4 caption"},{"comment":"The statement that the rho prediction is 'without any adjustable parameter' should be clarified: the parametrizations (20) and (21) have parameters fitted to total cross-section data, so the comparison in Fig. 3 is not a parameter-free prediction from first principles, although no additional parameter is introduced at this stage.","section":"Sec. 3, after Eq. (19)"}],"recommendation":"major_revision","confidential_remarks":"The central empirical observation—the constancy of Tbd from ISR to LHC—is solid and could justify publication if the paper is reframed. The main weakness is that the quantitative formulas (19) and (26) are presented as predictions although c0 is fitted and the odderon issue, acknowledged in the text, directly affects the same amplitude. A major revision should clearly separate the empirical scaling result from the model-dependent derivation and either extend the framework to C-odd amplitudes or explicitly quantify the sensitivity of Rbd to the rho discrepancy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is best read as a short phenomenological note, not a first-principles result. The empirical core—Tbd ≈ 1.355 constant from ISR to LHC and the dip position scaling as sigma_tot(s)|t|—is taken from the author's earlier paper with Baldenegro et al., and it's a nice regularity. What this paper adds is a crossing-based argument that ties rho and the bump-to-dip ratio Rbd to the energy derivative of the interaction radius. The derivation in Eqs. (14)–(18) is internally consistent, and Eq. (19) for rho is a clean, parameter-free expression once one picks a total cross section parameterization. The author is appropriately candid: he calls the analysis qualitative, flags the asymptotic Taylor expansion, and notes that low TOTEM rho points may signal an odderon.\n\nThe soft spots are real but proportionate. The most important is that Rbd, Eq. (26), is not really a prediction because c0 is chosen to match the data (0.012–0.013). Eq. (27) defines c0 through the unknown shape Phi, but the paper never computes it from a model; it's a fitting constant. The stress-test worry is on target here. If the true rho at 13 TeV is lower than Eq. (19) gives (say, because of an odderon), then Rbd = c0(1+rho^2)/rho^2 shifts by roughly -2 drho/rho, which is a large swing when rho goes from 0.135 to 0.10. The paper reproduces the Rbd data because c0 was chosen with the same rho that overshoots TOTEM; the tension is not resolved. The odderon-free assumption is acknowledged but not explored, and the ISR failure of the rho formula is mitigated only by an appeal to asymptotics.\n\nThat said, I don't think there's a load-bearing flaw. The geometric scaling observation is empirical and well supported; the crossing derivation is a reasonable interpretative framework, not a rigorous theorem. The paper is honest about what it can and cannot do. The literature coverage is adequate; the old dispersion-relation origin of Eq. (19) is credited.\n\nFor a referee: this deserves peer review, but you should ask for a discussion of how c0 relates to the shape of Phi, a sensitivity analysis of Rbd to rho, and error bars. As is, it's a worthwhile phenomenological note; with those additions it could be solid. I'd bring it to a reading group focused on elastic scattering phenomenology, though I wouldn't build on it until the c0 issue is clarified.","headline":"A transparent, modest phenomenological note: the empirical GS regularities are real, but the 'prediction' of the bump-to-dip ratio rests on a fitted constant and an odderon-free assumption.","tokens_in":9582,"tokens_out":6283,"would_cite":true,"duration_ms":59209,"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":"Geometric scaling survives at the LHC in the dip-bump region of elastic proton-proton scattering.","keywords":["elastic proton-proton scattering","geometric scaling","dip-bump structure","crossing symmetry","rho parameter","total cross section","Bessel-Fourier transform","LHC"],"falsifier":"Measure $\\rho$ at $\\sqrt{s} = 13$ TeV with total uncertainty below the gap between the value predicted by Eq. (19), computed from the measured slope $d\\sigma_{\\rm tot}/d\\ln s$, and the current low measurement; a deviation beyond the combined uncertainty, or an observed difference between proton-proton and proton-antiproton elastic cross sections in the dip region, would falsify the purely C-even geometric-scaling extraction.","tokens_in":8431,"feed_emoji":"📐","tokens_out":11265,"duration_ms":103512,"temperature":0.7,"pith_summary":"This paper sets out to show that geometric scaling, proposed and tested at 20-60 GeV in the 1970s, is still true at 13 TeV: elastic proton-proton scattering in the dip-bump region depends on energy only through the combination $\\tau = |t| R^2(s)$, with $R^2(s)$ proportional to $\\sigma_{\\rm tot}(s)$. The evidence is the constancy of the bump-to-dip position ratio, $1.355 \\pm 0.011$, across five decades of energy, which would not happen unless dips and bumps move with the same energy-dependent scale. The paper then combines this scaling assumption with crossing symmetry to identify the imaginary and real parts of the amplitude, producing a parameter-free formula for $\\rho$ and a related formula for the bump-to-dip cross-section ratio; both match collider data. The framework intentionally does not reproduce the energy dependence of the integrated elastic cross section, a failure the paper attributes to geometric-scaling violation at small momentum transfer.","feed_headline":"50-year-old scaling law survives at LHC in dip-bump region","feed_subtitle":"One scaling variable, the total cross section times momentum transfer, aligns the dip-bump pattern across five decades of energy.","key_machinery":"The load-bearing object is the scaling variable $\\tau = |t|R^2(s)$, with $R^2(s)$ proportional to $\\sigma_{\\rm tot}(s)$, together with the dimensionless profile $\\Phi(\\tau)$ whose Bessel-Fourier transform gives the imaginary part of the amplitude. The dip is the first zero of $\\Phi(\\tau)$; the bump is its first extremum. The argument runs through the crossing relation $\\widetilde{T}_{\\rm el}(-s,t) = \\widetilde{T}_{\\rm el}^{\\,*}(s,t)$, implemented by replacing $R^2(y)$ with $R^2(y - i\\pi/2)$ and keeping the first term in the Taylor expansion in the imaginary rapidity shift. This expansion is what converts the crossing constraint into a real part proportional to $dR^2/dy$, and it is the same mechanism that produces the formulas for $\\rho$ and $R_{\\rm bd}$. The machinery is deliberately qualitative, since the Taylor step is asymptotic and fails at the lowest ISR energies.","core_discovery":"The central discovery, as the paper states it, is that the dip-bump structure of elastic proton-proton scattering obeys geometric scaling from 20 GeV to 13 TeV with scaling variable $\\tau \\sim \\sigma_{\\rm tot}(s)|t|$, concretely through the constant position ratio $T_{\\rm bd} = |t_{\\rm bump}|/|t_{\\rm dip}| = 1.355 \\pm 0.011$. Applying crossing symmetry to the scaling ansatz $\\widetilde{T}_{\\rm el}(s,\\tau) = i s R^2(-is)\\, \\Phi(|t| R^2(-is))$ and expanding in the imaginary rapidity shift yields $\\mathrm{Re}\\,\\widetilde{T}_{\\rm el} = (\\pi/2)(dR^2/dy)\\, \\frac{d}{d\\tau}(\\tau\\Phi(\\tau))$. This gives $\\rho = (\\pi/2)(1/R^2)(dR^2/dy)$ and $R_{\\rm bd} = c_0 (1+\\rho^2)/\\rho^2$ with $c_0 \\simeq 0.012$-$0.013$, both of which track the data with no adjustable parameter. The same derivation overpredicts the energy growth of $\\sigma_{\\rm el}$, which the paper attributes not to a failure of geometric scaling in the region it tests but to the known violation of scaling at small $|t|$ that dominates the integrated elastic cross section.","pith_inferences":["If the low measured values of $\\rho$ at 13 TeV are taken at face value, a C-odd (odderon) contribution would enter the crossing relation at first order, so both the $\\rho$ and $R_{\\rm bd}$ formulas would need an odderon-dependent correction; fitting such a term would provide a clean two-component test.","The constancy of $T_{\\rm bd}$ could be tested with future 13.6 TeV data: a departure from $1.355$ would indicate that the scaling variable itself is energy-dependent beyond $\\sigma_{\\rm tot}|t|$.","The contrast between geometric scaling in the dip-bump region and its breakdown at small $|t|$ suggests that scaling is a property of the Bessel-zero structure of the Fourier-Bessel profile rather than of the full impact-parameter opacity; this could be checked with grey-disc models where the profile differs in shape but the zero structure is preserved.","A resummation of the Taylor expansion in the imaginary rapidity shift could extend the framework to ISR energies where the first-order result overshoots the data, possibly linking geometric scaling to Regge-pole intercepts."],"forward_implications":["Dips and bumps in $d\\sigma_{\\rm el}/dt$ from 20 GeV to 13 TeV should collapse onto a single curve when plotted against $\\tau = \\sigma_{\\rm tot}(s)|t|$.","The $\\rho$ parameter can be predicted from the measured energy slope of $\\sigma_{\\rm tot}$ with no free parameters, giving values that are consistent with collider measurements at LHC energies.","The bump-to-dip cross-section ratio $R_{\\rm bd}$ is not an independent number: it follows from $\\rho$ through $R_{\\rm bd} = c_0 (1+\\rho^2)/\\rho^2$, so the same $R^2(y)$ input that predicts $\\rho$ also predicts the observed saturation of $R_{\\rm bd}$ toward the LHC.","The failure of the same framework to reproduce the energy rise of $\\sigma_{\\rm el}$ implies that geometric scaling is violated outside the dip-bump region, most strongly at small $|t|$."],"supporting_citations":[{"why":"Introduces geometric scaling and the interaction-radius variable $b/R(s)$ that the momentum-space scaling variable $\\tau=|t|R^2(s)$ inherits.","marker":"[1]"},{"why":"Provides the elastic differential cross-section scaling law and the momentum-space ansatz that the paper extends to LHC energies.","marker":"[2]"},{"why":"Supplies the constant bump-to-dip position ratio $1.355 \\pm 0.011$ and the power-law dip fit used to identify the scaling variable.","marker":"[5]"},{"why":"Derives the crossing-symmetry expansion in the imaginary rapidity shift from which the real part of the amplitude is obtained.","marker":"[20]"},{"why":"Connects dips to zeros of the Bessel-Fourier profile and gives the ratio formula for bump-to-dip positions and values.","marker":"[22]"},{"why":"Provides one of the two total-cross-section parametrizations used to turn Eq. (19) into a parameter-free prediction for $\\rho$.","marker":"[24]"},{"why":"Provides the alternative two-power total-cross-section parametrization used to cross-check the $\\rho$ and $R_{\\rm bd}$ curves.","marker":"[26]"},{"why":"Supplies the low 13-TeV $\\rho$ measurement whose deviation motivates the odderon caveat on the crossing-only derivation.","marker":"[13]"}],"fun_headline_variants":["Geometric scaling holds at LHC for elastic pp","Bump-dip ratio stays constant from ISR to LHC","Scaling law from 1970s still valid at LHC","Elastic pp scattering preserves geometric scaling","Dip-bump pattern scales across five decades"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation presumes the proton-proton amplitude is purely symmetric under crossing (no odderon), so the real part is fixed by the scaling imaginary part alone; if the low measured values of $\\rho$ at 13 TeV are correct, that premise fails and the formulas for $\\rho$ and $R_{\\rm bd}$ become incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Geometric scaling holds at LHC for elastic pp","Bump-dip ratio stays constant from ISR to LHC","Scaling law from 1970s still valid at LHC","Elastic pp scattering preserves geometric scaling","Dip-bump pattern scales across five decades"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000357,"raw_usage":{"total_tokens":1943,"prompt_tokens":959,"completion_tokens":984,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":917}},"tokens_in":575,"tokens_out":984,"duration_ms":9512,"temperature":1.0,"reasoning_tokens":917,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:08:05.778480+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure $\\rho$ at $\\sqrt{s} = 13$ TeV with total uncertainty below the gap between the value predicted by Eq. (19), computed from the measured slope $d\\sigma_{\\rm tot}/d\\ln s$, and the current low measurement; a deviation beyond the combined uncertainty, or an observed difference between proton-proton and proton-antiproton elastic cross sections in the dip region, would falsify the purely C-even geometric-scaling extraction.","supporting_citations":[{"cited_title":"Geometric Scaling, Multiplicity Distributions and Cross-Sections,","cited_arxiv_id":null,"evidence_quote":"Introduces geometric scaling and the interaction-radius variable $b/R(s)$ that the momentum-space scaling variable $\\tau=|t|R^2(s)$ inherits."},{"cited_title":"Scaling law for the elastic differential cross-section in p p scattering from geometric scaling,","cited_arxiv_id":null,"evidence_quote":"Provides the elastic differential cross-section scaling law and the momentum-space ansatz that the paper extends to LHC energies."},{"cited_title":"Scaling laws of elastic proton-proton scattering differential cross sections,","cited_arxiv_id":null,"evidence_quote":"Supplies the constant bump-to-dip position ratio $1.355 \\pm 0.011$ and the power-law dip fit used to identify the scaling variable."},{"cited_title":"On the Real Part of a Geometrical Pomeron,","cited_arxiv_id":null,"evidence_quote":"Derives the crossing-symmetry expansion in the imaginary rapidity shift from which the real part of the amplitude is obtained."},{"cited_title":"Dips, Zeros and Large —t— Behavior of the Elastic Amplitude,","cited_arxiv_id":null,"evidence_quote":"Connects dips to zeros of the Bessel-Fourier profile and gives the ratio formula for bump-to-dip positions and values."},{"cited_title":"Hadronic scattering amplitudes: Medium-energy constraints on asymptotic behavior,","cited_arxiv_id":null,"evidence_quote":"Provides one of the two total-cross-section parametrizations used to turn Eq. (19) into a parameter-free prediction for $\\rho$."},{"cited_title":"Total cross-sections","cited_arxiv_id":null,"evidence_quote":"Provides the alternative two-power total-cross-section parametrization used to cross-check the $\\rho$ and $R_{\\rm bd}$ curves."},{"cited_title":"First determination of theρ parameter at √s = 13 TeV: probing the existence of a colourless C-odd three-gluon compound state,","cited_arxiv_id":null,"evidence_quote":"Supplies the low 13-TeV $\\rho$ measurement whose deviation motivates the odderon caveat on the crossing-only derivation."}],"review_version":1}