Universal properties of elastic pp cross section from the ISR to the LHC

Micha{\l} Prasza{\l}owicz

REVIEW 4 major objections 4 minor 2 cited by

Universal properties of elastic pp cross section from the ISR to the LHC

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Geometric scaling of elastic proton-proton scattering holds from 20 GeV to 13 TeV and fixes the real part of the amplitude.

desk verdict The empirical Tbd constancy is real and worth knowing, but the LHC 'universal' predictions are weaker than the abstract suggests—two of them fail against the paper's own numbers. read the letter →

arxiv 2505.11885 v1 pith:6WTAZI6D submitted 2025-05-17 hep-ph

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

classification hep-ph

keywords geometricscalingelasticppscatteringdip-bumpratiorhoparametercrossingsymmetrytotalcrosssectionISRLHC

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

The paper argues that the elastic proton-proton scattering amplitude in the diffractive dip-and-bump region depends on energy and momentum transfer only through the single scaling variable $\tau = R^2(s)|t|$, with $R^2(s) = \sigma_{\mathrm{tot}}(s)$. The empirical anchor is the constancy of the bump-to-dip position ratio, $T_{bd}=1.355\pm0.011$, from 20 GeV to 13 TeV. From this scaling plus crossing symmetry, the paper derives a prediction for $\rho$, the ratio of real to imaginary parts of the forward amplitude, and for the bump-to-dip cross-section ratio, and reports that both match ISR and LHC data. The stakes are that, if correct, geometric scaling is a valid approximate organizing principle for elastic high-energy scattering in the dip-bump region, and the real part of the amplitude is not an independent input but is fixed by the energy growth of the interaction radius.

What carries the argument

The machinery is the complex-scaling ansatz $\tilde T_{el}(s,\tau)=isR^2(-is)\Phi(|t|R^2(-is))$ together with the crossing-symmetry step that extracts real and imaginary parts: $\mathrm{Im}\,\tilde T_{el}=sR^2(y)\Phi(\tau)$ and $\mathrm{Re}\,\tilde T_{el}=s\frac{\pi}{2}\frac{dR^2}{dy}\frac{d}{d\tau}(\tau\Phi(\tau))$. This step converts the energy dependence of the radius $R^2(y)$ into a model-independent prediction for the real part of the amplitude, and hence for $\rho$, for the dip-bump cross-section ratio, and for the elastic cross section.

What would settle it

Compute the real part of the forward amplitude at $\sqrt{s}=13$ TeV by a standard dispersion integral using the same $\sigma_{\mathrm{tot}}$ parametrization as in the paper and compare with Eq. (7); if the two disagree by more than the experimental uncertainty, the analyticity trick is not equivalent to dispersion relations and the paper's predictions for $\rho$ collapse. Alternatively, a TOTEM measurement of $\rho$ at 13 TeV that deviates from $\frac{\pi}{2}\frac{d\ln\sigma_{\mathrm{tot}}}{dy}$ would directly falsify Eq. (7).

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Extended reading notes

Core claim

The central claim is that geometric scaling holds from the ISR to the LHC: in the dip-bump region the elastic amplitude is, up to normalization, a function of $\tau=R^2(s)|t|$ alone, not of $s$ and $t$ separately. With the ansatz $\tilde T_{el}(s,\tau)=isR^2(-is)\Phi(|t|R^2(-is))$ and the analyticity trick of the paper's reference [8], the paper identifies $\mathrm{Im}\,\tilde T_{el}=sR^2(y)\Phi(\tau)$ and $\mathrm{Re}\,\tilde T_{el}=s\frac{\pi}{2}\frac{dR^2}{dy}\frac{d}{d\tau}(\tau\Phi(\tau))$, where $y=\ln s$. This yields three predictions: $\rho = \frac{\pi}{2}\frac{1}{R^2}\frac{dR^2}{dy}$; a dip-bump cross-section ratio $R_{bd}=c_0(1+\rho^2)/\rho^2$ with one constant $c_0$; and an expression for $\sigma_{el}$. The paper reports that the $\rho$ prediction reproduces ISR and LHC data, and that $R_{bd}$ is fitted with $c_0\approx0.012\text{--}0.013$, while the $\sigma_{el}$ prediction fails at LHC because geometric scaling breaks down at small $t$.

Load-bearing premise

Everything rests on the claim that the mathematical step identifying the real part of the amplitude from the energy derivative of the scaling radius is exactly what a dispersion relation would produce; if that step is only approximate, the $\rho$ and $R_{bd}$ predictions do not follow from geometric scaling.

Editorial extensions

If this is right

Dip and bump positions scale as $1/R^2(s)$, so any future collider energy where this scaling holds will have the same $T_{bd}=1.355$ ratio.
The real part of the amplitude is nonzero at the dip, where the imaginary part vanishes; the dip cross section is therefore set by $\rho$, making the dip-bump ratio $R_{bd}$ a clean probe of the real part.
The $\rho$ parameter is entirely determined by the logarithmic slope of $\sigma_{\mathrm{tot}}$; a precise $\rho$ measurement at any energy is a direct test of the analyticity step.
The elastic cross-section prediction fails at LHC energies, which means geometric scaling is not global; it is valid only in the $t$-range away from very small $|t|$.

Reading between the lines

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

A natural extension is to apply the same scaling ansatz to proton-antiproton elastic scattering; if the odderon contribution is real, the crossing-symmetry step would need modification, and the equality of $\rho$ predictions between $pp$ and $p\bar p$ could discriminate.
The equivalence of the analyticity trick to dispersion relations is asserted but not demonstrated; a direct numerical evaluation of a subtracted dispersion integral for $\sigma_{\mathrm{tot}}$ at LHC energies would test whether the extracted real part is quantitatively faithful.
If $T_{bd}$ is measured at a future high-energy collider and remains $1.355$, that supports the scaling variable $\tau=\sigma_{\mathrm{tot}}|t|$; if it drifts, the breaking point would mark where new dynamics enters.
The paper uses $\sigma_{\mathrm{tot}}$ for $R^2$; one could test whether using a different radius, say extracted from the dip position itself, improves the $\rho$ prediction or changes $c_0$.

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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

Quick take: the paper has one solid empirical fact — the dip-to-bump position ratio Tbd = 1.355 ± 0.011 is constant from the ISR to the LHC. That is a clean direct ratio from data, with a stated uncertainty, and it is the real contribution. But the "universal" predictions built on it are less secure than the abstract promises. The rho formula overshoots TOTEM's 13 TeV value by about 30%, which the author attributes to an odderon; the Rbd formula with the paper's own c0 and rho gives roughly 1.3, not the ~1.8 saturation quoted in Section 2.

Novelty is modest. The constant Tbd already appears in Ref. [2] with overlapping authorship, the LHC geometric-scaling argument is in Ref. [1] by the same author, and the rho construction follows Dias de Deus's 1975 paper. What is new is the explicit numerical comparison of these known relations against the COMPETE and DL parametrizations, plus the observation that GS at the LHC holds only in a narrower t window around the dip-bump region. That packaging is useful, but this reads more like a review-with-illustrations than a new result.

To give credit where it is due: the text is honestly self-aware about its failures. It concedes that GS breaks down at small t, that the LHC integrated elastic cross section is not reproduced, and that the last two TOTEM rho points do not fit Eq. (7). Showing the failures rather than hiding them is a point in the author's favor.

The soft spots are proportional to how much weight they carry. The derivation of Eq. (6) is presented as "the trick of Ref. [8]" with no derivation; the stress-test note is right that this is actually the leading-order derivative dispersion relation, so the gap is a presentation issue, not a load-bearing hole. The quantitative problem is at LHC energies, where epsilon = d ln sigma_tot/d ln s is around 0.09–0.17 and the O(epsilon^2) corrections are not negligible. The internal inconsistency in Eq. (8) — giving ~1.3 instead of ~1.8 — is real and should be fixed. And calling the rho comparison "well reproduced" overstates what the data actually show.

None of this undercuts the empirical Tbd constancy. But "universal properties" is too strong a title when the LHC part of the prediction fails on the paper's own numbers.

Who is this for: hadronic phenomenology people working on elastic scattering, geometric scaling, and the odderon. I would bring it to a phenomenology reading group as a discussion piece, but I would cite Ref. [2] for the Tbd constant rather than this paper.

Recommendation: yes, send it to peer review. A serious referee should require a proper derivation of Eq. (6), corrected Rbd numbers, and toned-down language about LHC agreement. These are addressable requests, not grounds for rejection.

Referee Report

4 major / 4 minor

Summary. The paper argues that the ratio Tbd = tbump/tdip of the elastic pp differential cross section is energy independent, Tbd = 1.355 ± 0.011 from the ISR to the LHC, and uses this observation to promote geometric scaling in the variable τ = R^2(s)|t|. Assuming crossing symmetry and the complex-scaling ansatz of Eq. (5), the author identifies real and imaginary parts of the amplitude via Eq. (6), and derives predictions for the rho parameter (7), for the bump-to-dip ratio of cross-section values Rbd (8), and for the elastic cross section σel (9). With R^2(s) = σtot(s) and two analytic parametrizations of σtot, these predictions are compared with data, and the paper concludes that geometric scaling explains the main properties of elastic pp scattering despite violations at small t and the odderon-related behavior of the last TOTEM rho points.

Significance. The empirical constancy of Tbd is a clean, falsifiable observation and, if correct, would be a useful organizing principle for elastic pp scattering over five decades in energy. The paper deserves credit for stating the direct empirical input explicitly and for identifying which quantities are fitted and which are derived. However, the derived predictions are not parameter-free: Eq. (7) requires the choice R^2 = σtot, Eq. (8) contains the fitted constant c0, Eq. (9) depends on an unknown c1, and the analyticity step Eq. (6) is imported from Ref. [8] without derivation. As discussed below, the quantitative support in Section 3 has internal tensions at LHC energies, so the paper's significance would be substantially increased by a careful revision that quantifies the approximations and reconciles the model with the LHC data.

major comments (4)

[Eq. (6), Section 2] The derivation of Eq. (6) is load-bearing: all three predictions, Eqs. (7), (8), and (9), follow from it, yet the manuscript only cites "the trick of Ref. [8]" and asserts equivalence to dispersion relations. Please provide a self-contained derivation or state the approximation explicitly. As written, Eq. (6) is the leading-order derivative dispersion relation and receives relative corrections of order ε^2, where ε = d ln R^2/d ln s, and at LHC energies ε is not numerically negligible.
[Eq. (8), Section 3] There is an internal inconsistency in the Rbd prediction at LHC energies. With the fitted c0 = 0.012–0.013, Eq. (8) gives Rbd ≈ 1.26–1.37 for the TOTEM value rho = 0.098 ± 0.003, and Rbd ≈ 0.74–0.81 for the DL parametrization value rho ≈ 0.127. These values are far from the quoted LHC saturation value Rbd ≈ 1.8 in Section 1 and the right panel of Fig. 2. The paper does not address this discrepancy; it should either explain why Eq. (8) is not expected to describe the LHC values, allow c0 to depend on energy, or present the model curve together with the data and an error estimate.
[Eq. (7), Section 3] For the DL parametrization used in Fig. 1, Eq. (7) asymptotes to (π/2) × 0.0808 = 0.127 at 13 TeV, overshooting the TOTEM value rho = 0.098 ± 0.003 by roughly 30%. The text acknowledges this by attributing the last two TOTEM points to the odderon, but this means Eq. (7) does not reproduce those data points. Please quantify the mismatch and state clearly that the LHC rho points are a failure of the prediction rather than a success, or provide a modified form of Eq. (7) that includes the odderon contribution.
[Eqs. (7)–(9), Section 2 and 3] The comparisons in Section 3 are not parameter-free tests of geometric scaling. Eq. (7) uses R^2(s) = σtot(s) chosen by hand from parametrizations fitted to total cross-section data; Eq. (8) contains c0 fitted to the Rbd data being compared; Eq. (9) contains an unknown c1 and is only discussed qualitatively. The manuscript should state explicitly, for each of the three equations, which aspects are predicted and which are fitted, so that the reader can assess the strength of the evidence.

minor comments (4)

[Section 1] The text says "Changing variables in (2)", but the relevant integral definitions appear in Eq. (1), not Eq. (2).
[Section 1, Eq. (3)] The notation switches between s and W = √s; Eq. (3) uses W, while the scaling variable τ = R^2(s)|t| uses s. Please define the argument of R^2 consistently.
[Section 1] The sentence "saturation at ~1.8 at the LHC (see Fig. 1)" refers to Rbd, but Fig. 1 shows total cross sections; the relevant plot is Fig. 2.
[Section 3, Fig. 2] The right panel of Fig. 2 cites Ref. [5] for the Rbd data, but Ref. [5] includes both TOTEM and D0 data; please specify which data points are used in the comparison.

Circularity Check

1 steps flagged · score 4.0 of 10

Partially circular: Eq. (8) is advertised as a prediction, but its normalization is fixed by the adjustable parameter c0 fit to the same Rbd data being compared; the geometric-scaling premise and the rho cross-check retain independent empirical content.

fitted input called prediction [Section 2 (Eq. 8) and Section 3, right-panel paragraph: 'In the right panel of Fig. 2 we plot Rbd ratio computed according to (8) with one adjustable parameter c0 = 0.012÷ 0.013. Data are from Ref.]
"The second one is a prediction for Rbd. ... In the right panel of Fig. 2 we plot Rbd ratio computed according to (8) with one adjustable parameter c0 = 0.012÷ 0.013. Data are from Ref. [5]."

Equation (8) is called a prediction, but the constant c0 = Phi^2(tau_bump)/(tau_dip dPhi/dtau(tau_dip))^2 is not fixed by the theory. In the phenomenology section c0 is adjusted to the range 0.012-0.013 and the resulting curve is compared with the same Rbd data, from Ref. [5], that were used to set it. Consequently the normalization of the Rbd comparison is fitted rather than predicted; only the energy dependence entering through rho(y) is independently tested. The plotted agreement is therefore partly a one-parameter fit to the data it claims to predict.

full rationale

The paper's central empirical input, Tbd = 1.355 +/- 0.011, is a measured ratio taken from data (Ref. [2]) and is not derived from the model, so establishing geometric scaling is not circular. The analyticity shortcut in Eqs. (5)-(6) is imported from the older, independent Ref. [8] and is not a self-citation chain. Equation (7) for rho is a genuine cross-check insofar as R^2(s) is obtained from sigma_tot parametrizations; the paper even concedes that the last TOTEM rho points deviate, which shows falsifiability. Equation (9) is explicitly not used because c1 is unknown. The only clear circular element is Eq. (8): the 'prediction for Rbd' is normalized by a constant c0 fitted to the same Rbd data with which it is compared. That is a partial fitted-input-called-prediction, but the overall geometric-scaling claim and the rho comparison are not reduced to fit, so the paper is only moderately circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The load-bearing inputs are the geometric scaling ansatz, a dispersion-relation shortcut, and the identification R^2 = sigma_tot. Two constants remain: c0 is fitted to data and c1 is unknown. No new entities are introduced.

free parameters (3)

c0 = 0.012 to 0.013
Normalization of the Rbd prediction in Eq. (8), fitted to the same bump/dip value data it is compared with in Fig. 2.
R^2(s) = sigma_tot(s) = COMPETE and Donnachie-Landshoff parametrizations
The scaling radius is chosen by hand to be the total pp cross section in Sec. 3; all numerical rho predictions inherit this choice.
c1 = not determined
Constant in Eq. (9) controlling the real-part correction to sigma_el; never evaluated, so the elastic cross section prediction is incomplete.

assumptions (4)

domain assumption Geometric scaling ansatz: T~el(s,t) = i s R^2(-is) Phi(|t| R^2(-is)), Eq. (5), with Phi energy independent.
Starting model inherited from Refs. [1,8]; the paper tests its consequences rather than deriving it.
domain assumption The Ref. [8] trick identifies imaginary and real parts of the amplitude as in Eq. (6), with the real part proportional to dR^2/dy times d/dtau(tau Phi).
Assumed equivalent to dispersion relations; no proof or validity conditions are given in this paper, and all three predictions depend on it.
domain assumption R^2(s) may be replaced by sigma_tot(s) for numerical comparison.
A choice made in Sec. 3 without an independent argument; it sets the scale of tau and drives the rho prediction.
domain assumption The real part of the elastic amplitude can be neglected in the integrated cross sections of Eq. (2) because rho is small.
The paper says smallness of rho justifies this, but rho grows with energy and the LHC integrated cross sections themselves violate the naive scaling.

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Cite this review

Pith. "Pith review of Universal properties of elastic pp cross section from the ISR to the LHC." pith.science (2026). https://pith.science/paper/6WTAZI6D

@misc{pith2026250511885,
  author       = {Pith},
  title        = {Pith review of: Universal properties of elastic pp cross section from the ISR to the LHC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WTAZI6D}},
  note         = {Machine review of arXiv:2505.11885}
}

abstract

We explore the phenomenology of the property that the ratio of bump to dip {\em positions} of the elastic differential $pp$ cross section is constant over the energy range from the ISR to the LHC. We review the old idea of geometric scaling at the ISR and argue that it also holds at the LHC. We discuss its consequences for the $\rho$ parameter and for the ratio of bump to dip cross section {\em values}.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

Geometric scaling in elastic $pp$ collisions
hep-ph 2026-07 conditional novelty 2.0 of 10

Elastic pp scattering shows a universal bump-to-dip position ratio T_bd=1.355 from ISR to LHC, and analyticity then yields the rho parameter and bump-to-dip ratio.
Theoretical Summary: Moriond QCD and High-Energy Interactions 2025
hep-ph 2025-06 unverdicted

A conference summary that compiles the main theory results presented at Moriond QCD 2025, spanning hard scattering, precision QCD, flavour, strong coupling, lattice, heavy-ion, and BSM physics.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [2]

Baldenegro, M

C. Baldenegro, M. Praszalowicz, C. Royon and A. M. Stasto. Scaling laws of elastic proton-proton scattering differential cross sections. Phys. Lett. B , 856:138960, 2024

work page 2024

[2] [1]

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

M. Prasza lowicz. Geometric scaling of elastic pp cross section at the LHC . [arXiv:2504.18841 [hep-ph]]

work page Pith review arXiv

[3] [8]

Dias de Deus

J. Dias de Deus. On the Real Part of a Geometrical Pomeron. Nuovo Cim. A 28:114, 1975

work page 1975

[4] [3]

Dias De Deus

J. Dias De Deus. Geometric Scaling, Multiplicity Distributions and Cross-Sections. Nucl. Phys. B 59:231, 1973

work page 1973

[5] [4]

A. J. Buras and J. Dias de Deus, Scaling law for the elastic differential cross-section in pp scattering from geometric scaling, Nucl. Phys. B 71:481, 1974

work page 1974

[6] [5]

V. M. Abazov et al. (TOTEM and D0), Odderon Exchange from Elastic Scattering Differ- ences betweenpp andp¯p Data at 1.96 TeV and from pp Forward Scattering Measurements, Phys. Rev. Lett. 127:062003, 2021

work page 2021

[7] [6]

Baldenegro, C

C. Baldenegro, C. Royon and A. M. Stasto. Scaling properties of elastic proton-proton scattering at LHC energies. Phys. Lett. B 830:137141, 2022

work page 2022

[8] [7]

Dias de Deus and P

J. Dias de Deus and P. Kroll. Dips, Zeros and Large|t| Behavior of the Elastic Amplitude. Acta Phys. Pol. B 9:157, 1978

work page 1978