Geometric scaling of elastic $pp$ cross section at the LHC

Micha{\l} Prasza{\l}owicz

REVIEW 3 major objections 4 minor 1 cited by

Geometric scaling of elastic $pp$ cross section at the LHC

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Geometric scaling survives at the LHC in the dip-bump region of elastic proton-proton scattering.

desk verdict 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. read the letter →

arxiv 2504.18841 v2 pith:JJ6GL4PK submitted 2025-04-26 hep-ph

Micha{l} Prasza{l}owicz This is my paper · ORCID

classification hep-ph

keywords elasticproton-protonscatteringgeometricscalingdip-bumpstructurecrossingsymmetryrhoparametertotalcrosssectionBessel-FouriertransformLHC

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $|t|$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The 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.

The 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.

That 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.

For 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Sec. 3, Eq. (26) and Fig. 4] 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.
[Sec. 3, Eqs. (19) and (26)] 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.
[Sec. 2, Eqs. (16)-(17) and Fig. 3] 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.

minor comments (4)

[Sec. 2, Eq. (23)] 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).
[Throughout] There are several typographical errors, including 'gown' for 'down' in Section 4, 'tis' for 'this', and 'limitted' for 'limited'. These should be corrected.
[Fig. 4 caption] 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.
[Sec. 3, after Eq. (19)] 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.

Circularity Check

1 steps flagged · score 6.0 of 10

The Rbd 'prediction' in Eq. (26) uses a constant c0 that is fitted to the same data it is said to reproduce, so the agreement in Fig. 4 is partly forced.

fitted input called prediction [Section 3, Eqs. (25)-(27) and Fig. 4 (text following Eq. 27)]
"Hence, we have a one parameter prediction for the ratio Rbd(s) = dσ/dt(tbump)/dσ/dt(tdip) = c0 (1+ρ^2)/ρ^2, where constant c0 is given in terms of function Φ ... In Fig. 4 we plot ratio (26) for two parametrizations (20) and (21) with c0 = 0.012÷ 0.013. One can see that the ratio data are well reproduced by both parametrizations."

Eq. (27) defines c0 through an unspecified function Φ, but no model for Φ is supplied. The number c0 = 0.012–0.013 is chosen after the fact so that the curves in Fig. 4 line up with the same Rbd data that Eq. (26) is said to reproduce. Therefore the normalization of Eq. (26) is a fitted input, not a prediction. The energy-dependent shape comes from the derived ρ(y), but the absolute level that yields agreement is imposed by c0. The black-disc Φ shown in Fig. 1 is not used to fix c0, so Eq. (27) does not supply the value; the 'one parameter prediction' is thus partly a consistency check with a constant fitted to the data.

full rationale

The central crossing derivation (Eqs. 13-19) is self-contained and follows published work by Dias de Deus and Kroll; no circular step is involved there. The 'absolute prediction' for ρ is a known dispersion-relation result evaluated with global parametrizations, making Fig. 3 a consistency check rather than an independent test, but not a self-definitional circle. The empirical scaling Tbd = 1.355 from Ref. [5] is imported from a prior publication with overlapping authorship, but it is a data-analysis result and is not used to derive itself, so I do not count it as load-bearing circularity. The actual circular element is Eq. (26): with Φ left unspecified, c0 is a free constant, and the paper sets it to 0.012–0.013 to match the very Rbd data it claims to reproduce. Thus the bump-to-dip cross-section ratio, one of the two quantitative outputs advertised in the abstract, is partly determined by a fitted normalization. This warrants a partial-circularity score of 6.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central result rests on the GS ansatz for the amplitude, the crossing-based extraction of Re T, the first-order expansion in the rapidity shift, and the identification of R^2 with sigma_tot. The only new free parameter introduced is c0, which is fitted to the Rbd data; all other inputs are established parametrizations or inputs from the author's prior work.

free parameters (1)

c0 = 0.012 to 0.013
Normalization constant in Eq. (27) and (26) for the bump-to-dip ratio Rbd; chosen by hand to match the data in Fig. 4, not computed from the explicit form of Phi(tau).

assumptions (5)

domain assumption The elastic amplitude is purely C-even (no odderon contribution); crossing relation (13) alone determines the real part.
Invoked in Sections 2 and 3 to derive Re T from Im T via crossing; the author notes that TOTEM rho deviations may signal an odderon, which is beyond the scope.
ad hoc to paper First-order Taylor expansion of R^2(y - i*pi/2) and Phi with respect to the imaginary rapidity shift is valid, Eqs. (16)-(17).
The derivation of rho and Rbd keeps only linear terms; the paper states the expansion is asymptotic and overshoots ISR data.
domain assumption Geometric scaling ansatz (8)/(14): the amplitude depends on the single variable tau = |t| R^2(s), with R^2 proportional to sigma_tot(s).
This is the central modeling assumption taken from Refs. [1,2]; it is tested only in the dip-bump region.
domain assumption The total cross section parametrizations (PDG 2010 and Donnachie-Landshoff) describe the world data well.
R^2 is identified with sigma_tot; the curves for rho and Rbd inherit these parametrizations. The DL parametrization is known to undershoot LHC total cross sections.
standard math Textbook normalization of the elastic amplitude and cross sections, Eqs. (1), (6), (7).
Standard definitions used without proof, forming the basis for the GS expressions.

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Pith. "Pith review of Geometric scaling of elastic $pp$ cross section at the LHC." pith.science (2026). https://pith.science/paper/JJ6GL4PK

@misc{pith2026250418841,
  author       = {Pith},
  title        = {Pith review of: Geometric scaling of elastic $pp$ cross section at the LHC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJ6GL4PK}},
  note         = {Machine review of arXiv:2504.18841}
}

abstract

We show that geometric scaling conjectured and observed at the ISR more than 50 years ago, still holds at the LHC. We discuss regularities of the dip-bump structures of the differential elastic cross sections, emphasizing the fact that the ratio of bump to dip positions is constant from the ISR to the LHC. Applying crossing and analyticity we identify imaginary and real parts of the scattering amplitude and compute the $\rho$ parameter and the ratio of bump to dip values of the differential $pp$ cross sections. We also discuss the energy dependence of the total elastic cross section and the violation of geometrical scaling outside the dip-bump region at the LHC.

Figures

Figures reproduced from arXiv: 2504.18841 by the authors.

**Figure 1.** Left panel: function Φ(τ) from Eq. (11) – solid red line, and d(τΦ(τ))/dτ – dashed blue. Right panel: Contributions of imaginary (solid red) and real (dashed blue) parts of the scattering amplitude to the elastic cross section. For the real part the ρ parameter is 0.15. Expressing R2 in terms of rapidity y = ln s rather than s, and expanding, we get [20] R 2 (−is) → R 2 y − i π 2 ≃ R 2 (y) − i π 2 dR2 (y) dy . (… view at source ↗

**Figure 2.** Total pp cross-section in mb as a function of √ s in GeV. Points at small √ s < 100 GeV are from the ISR, points above 1 TeV are from the LHC (small error bars) and from cosmic rays. Solid magenta line corresponds to the PDG parametrization (20) and brown dashed line to (21). where Z = 35.45 mb, C = 0.308 mb, Y1 = 42.53 mb, Y2 = 33.34 mb, s0 = 28.94 GeV2 , s1 = 1 GeV2 and η1 = 0.458, η2 = 0.545. The second one is fr… view at source ↗

**Figure 3.** Parameter ρ (19) as a function of √ s in GeV. Points at small √ s < 100 GeV are from the ISR, points above 1 TeV are from the LHC. Solid magenta line corresponds to the PDG parametrization (20) and brown dashed line to (21). Therefore dσ dt [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Ratio Rbd as a function of √ s in GeV. Data: brown triangles from Ref. [5], blue circles from Ref. [21]. Theoretical curves (26) as in Figs. 2 and 3. At the ISR the ρ parameter is very small and, despite the fact that it rapidly rises with energy, its influence on σel …

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Universal properties of elastic pp cross section from the ISR to the LHC
hep-ph 2025-05 conditional novelty 4.0 of 10

The dip-bump positions of elastic proton-proton scattering are aligned by geometric scaling from ISR to LHC, and the same scaling predicts the rho parameter when the radius is identified with the total cross section.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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