Geometric scaling in elastic $pp$ collisions

Micha{\l} Prasza{\l}owicz

REVIEW 4 major objections 5 minor 15 references

Geometric scaling in elastic $pp$ collisions

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Elastic proton-proton cross-sections exhibit geometric scaling from ISR to LHC via a constant ratio of bump to dip positions, T_bd = 1.355 ± 0.011.

desk verdict A clear proceedings summary of already-published results; the T_bd constancy is real but does not by itself establish full geometric scaling at the LHC. read the letter →

arxiv 2607.16182 v1 pith:TJW3N4MI submitted 2026-07-17 hep-ph

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

classification hep-ph

keywords geometricscalingelasticproton-protonscatteringdip-bumpstructurerhoparametercrossingsymmetrytotalcross-sectionLHCISR

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

Geometric scaling in elastic proton-proton scattering was conjectured at the ISR in the 1970s and seemed to fail at the LHC because integrated cross-sections grow with different powers of energy. This paper argues the scaling survives inside a narrow kinematic window: the positions of the diffractive dip and bump obey |t_dip| and |t_bump| = 1.355 times |t_dip| with the same energy dependence at every measured energy from 23 GeV to 13 TeV. Because the positions scale as 1/sigma_tot(s), one energy-dependent radius aligns all dip and bump data. Using crossing symmetry and the optical theorem, the paper identifies the real part of the elastic amplitude and derives parameter-free predictions for the rho parameter and a one-parameter prediction for the bump-to-dip cross-section ratio, both in agreement with data. This matters because it shows a simple, intuitive picture still organizes high-energy pp scattering where it was thought to break down.

What carries the argument

The paper's load-bearing object is the scaling variable tau = |t| sigma_tot(s) (equivalently R^2(s) = sigma_tot(s)), through which the universal function Phi(tau) controls dip and bump positions. The constancy of T_bd = t_bump/t_dip is the empirical hook. The argument's engine is the crossing relation T_el(u,t) ≈ T*_el(s,t) combined with the analytic expansion -i s = e^{y - i pi/2}; expanding R^2(-i s) and Phi(|t| R^2(-i s)) to first order splits the amplitude into imaginary and real parts, making predictions (22), (25), and (27) possible.

What would settle it

Precision measurement of rho at sqrt(s) = 13 TeV would settle the matter: if rho continues its rapid decrease with energy, the crossing-even assumption is wrong and equations (18) and (22) fail. Alternatively, measuring T_bd at a new energy, such as 13.6 TeV or a future collider, and finding a value outside 1.355 ± 0.011 would falsify geometric scaling in the dip-bump region.

Watch

Extended reading notes

Core claim

The central discovery is that the ratio T_bd = t_bump/t_dip is 1.355 ± 0.011, constant across nine data sets spanning 23 GeV to 13 TeV. This implies |t_dip| and |t_bump| both scale as tau_dip/sigma_tot(s) and tau_bump/sigma_tot(s), so the cross-section depends on tau = |t| sigma_tot(s) in this region. The paper then imposes the crossing relation T_el(u,t) ≈ T*_el(s,t) and expands around -i s = exp(y - i pi/2), obtaining Im T = s R^2 Phi(tau), Re T = s (pi/2)(dR^2/dy) d(tau Phi)/d(tau). From these it computes rho = (pi/2)(1/R^2)(dR^2/dy) and R_bd = c_0 (1 + rho^2)/rho^2, and shows both match ISR and LHC data.

Load-bearing premise

The derivation requires the amplitude to be crossing-even, T_el(u,t) ≈ T*_el(s,t), with no Odderon contribution; this assumption is directly challenged by the rapid drop of the measured rho values at 13 TeV, which the paper itself attributes to the Odderon.

Editorial extensions

If this is right

If T_bd is constant, dip and bump positions at all energies are determined by a single energy-dependent radius R^2(s) = sigma_tot(s); future measurements at higher energies should see |t_dip| and |t_bump| continue as 1/sigma_tot(s).
The rho parameter is predicted without free parameters from the energy dependence of sigma_tot(s); both ISR and LHC data are reproduced, except the last high-energy points whose rapid drop is attributed to the Odderon.
The bump-to-dip cross-section ratio R_bd is predicted as c_0(1 + rho^2)/rho^2 with one free constant; data follow this form.
The total elastic cross-section differs from sigma_tot by the factor 1 + c_1 rho^2(s), which is negligible at ISR and LHC; the observed LHC discrepancy is blamed on GS violation at small |t| outside the dip-bump region.
Geometric scaling does not hold globally at the LHC; its validity is restricted to the dip-bump region.

Reading between the lines

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

If the constancy of T_bd is exact, the shape function Phi(tau) is universal and energy independent; one could test this by predicting the position of the second dip (third zero) of the amplitude at LHC energies.
The crossing-even assumption (Eq. 13) is the soft spot; a precise rho measurement at 13 TeV and beyond would decide whether the predicted real part holds or whether an Odderon component is needed.
The same scaling logic could be applied to proton-antiproton scattering; a comparison of pp and ppbar dip-bump patterns would directly quantify the Odderon contribution.
Because the real part is built from the energy derivative of R^2, the method connects the rho parameter's energy rise at LHC to the total cross-section's growth; if future data flatten sigma_tot, rho should flatten too.

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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

This is a proceedings paper, and it reads like one: it summarizes the author's recent work (Refs. [2,3]) rather than presenting new results. That is not a flaw in itself, but it matters for how you weigh the claims. What is genuinely nice here is the empirical regularity: T_bd, the ratio of bump to dip positions, is constant at 1.355 ± 0.011 from ISR to 13 TeV. That is a compact, useful observation, and the fits look clean. The derivation of rho from the energy slope of sigma_tot via crossing is a neat trick, and the paper is honest about where the approach fails: it does not reproduce the energy growth of sigma_el at the LHC, and it flags the small-t region as violating GS.

The soft spots are real but not evenly soft. The biggest one is the title-level claim that geometric scaling "still holds at the LHC." What actually holds is position scaling: the dips and bumps align when you scale t by sigma_tot. The paper itself says the cross-section values at LHC can only be "approximately superimposed by a different function of s," which means Eq. (12) — the universal curve for sigma_tot^2 dsigma/dt — does not hold. So T_bd constancy is a kinematic regularity, not evidence for full GS. The stress-test note gets this right.

Second, the rho prediction is genuinely parameter-free given a sigma_tot parametrization, but it visibly misses the last two TOTEM points, which turn down. The paper attributes that to the odderon. That is a direct conflict with the crossing-even assumption (13) used to derive Eq. (22). You cannot invoke odderon to explain away a failure of a derivation that assumed no odderon. This is a load-bearing tension, not a minor caveat.

Third, the R_bd "prediction" has c0 as a free parameter fit to data, so it is a fit. The sigma_el prediction has c1 not computed, so it is incomplete. The paper admits both, but the abstract still talks about predictions. The prose is a bit more careful than the abstract.

Who gets value? A phenomenologist working on elastic pp scattering, or someone who wants a quick summary of the Baldenegro-Praszalowicz-Royon-Stasto scaling laws. It is not the place to go for new derivations. The citation pattern is fine—it cites the original 1970s GS papers and the recent odderon papers.

I would send this to a referee, not desk-reject it, because the empirical T_bd constancy and the rho discrepancy deserve scrutiny. But the referee would need to push the authors to stop calling it geometric scaling at the LHC when it is, at best, position scaling, and to address the odderon conflict head-on.

Referee Report

4 major / 5 minor

Summary. The paper argues that geometric scaling (GS) holds in elastic pp scattering at LHC energies, at least in the dip-bump region. The main empirical observation is that the ratio T_bd = |t_bump|/|t_dip| is constant, T_bd = 1.355 ± 0.011, from ISR (23 GeV) to LHC (13 TeV). The author then assumes a purely imaginary scaling amplitude with R^2(s)=σ_tot(s), applies crossing symmetry and analyticity to obtain the real part of the amplitude, and derives expressions for the ρ parameter, the bump-to-dip cross-section ratio R_bd, and the elastic cross-section σ_el. The paper concludes that the main properties of total and differential cross sections can be explained from GS, while acknowledging that GS is violated outside the dip-bump region at LHC energies.

Significance. The empirical constancy of T_bd, if robust, is a simple and striking regularity that deserves attention; the paper correctly emphasizes that it is largely unexpected and provides a quantitative fit. The derivation of ρ from the energy dependence of σ_tot is parameter-free once R^2=σ_tot is assumed, and the comparison with low-energy data is a genuine consistency check. However, the paper's central claim that GS 'still holds at the LHC' is only partially supported: the evidence is limited to the scaling of dip/bump positions, not of cross-section values, and the theoretical extensions involve a fitted constant c0 and an incomplete evaluation of c1. The most serious issue is that the crossing-even assumption used to derive the real part is contradicted by the TOTEM ρ measurement at 13 TeV, which the paper itself attributes to the Odderon. These gaps prevent the paper from establishing GS as a full dynamical statement at LHC energies, although the empirical position-scaling regularity remains valuable.

major comments (4)

[Section 2, Eq. (12) and Fig. 2] The inference from T_bd constancy to GS at the LHC is incomplete. GS as defined by Eq. (8) requires the amplitude to depend only on τ=-t R^2(s); for R^2=σ_tot this gives the universal curve σ_tot^2 dσ/dt (Eq. 12). The paper shows this works at ISR (Fig. 2), but explicitly states that at LHC 'the cross-section values can be approximately superimposed by a different function of s'. Thus σ_tot^2 dσ/dt is not universal at LHC, so the amplitude is not a function of τ alone. T_bd constancy only establishes that |t_dip| and |t_bump| share the same energy dependence; it does not establish GS of the amplitude. The abstract's claim that 'geometric scaling still holds at the LHC' therefore needs to be qualified as position scaling only, or supported by an explicit test of value scaling in the dip-bump region.
[Section 4, Eqs. (25)-(26)] Equation (25) is called a 'one parameter prediction' for R_bd, but c0 is explicitly treated as a free parameter and fitted (c0 ≃ 0.03). Since c0 is defined in Eq. (26) in terms of the unknown shape function Φ and is not computed from an independent input, Eq. (25) is a fit, not a prediction. The agreement shown in Fig. 6 therefore tests only the functional form of the ρ-dependence, not the GS hypothesis. This should be stated transparently, and the term 'prediction' should be avoided unless c0 is pinned down or a different prediction is made.
[Section 3, Eq. (13), and Section 4, Eq. (22)] The derivation of the real part and the ρ parameter rests entirely on the crossing-even relation T_el(-s,t)=T*_el(s,t). The paper itself notes that the TOTEM ρ measurements at 13 TeV decrease rapidly with energy and attributes this to the Odderon. A nonzero Odderon contribution violates Eq. (13) at that energy. Therefore Eqs. (18) and (22) are not valid at the very LHC energies where the paper applies them. The low-energy agreement is interesting, but the 13 TeV discrepancy is not an external nuisance: it is a direct failure of the crossing-even assumption. The paper should either restrict the theoretical claims to energies where crossing-even is justified, or incorporate a C-odd term and show how the predictions change.
[Section 4, Eqs. (27)-(28)] The σ_el prediction is incomplete. The factor 1+c1 ρ^2 is written down, but c1 is not computed, so the energy dependence of σ_el is not actually predicted. Moreover, Eq. (27) integrates over all τ, which requires GS to hold everywhere in t; the paper later states that GS fails outside the dip-bump region at the LHC. Consequently, the statement that the LHC σ_el slope can be attributed to GS violation is reasonable but untested by Eq. (27). Either c1 must be evaluated for a specific Φ or this section should be presented as a qualitative discussion rather than a prediction.

minor comments (5)

[General] Typo: 'analicity' should be 'analyticity'. Also 'attributed the odderon' should be 'attributed to the Odderon'.
[Section 2, Fig. 2 caption] The caption says R^2=σ_inel was used, while the text uses R^2=σ_tot elsewhere. Clarify why the inelastic cross-section is used in the ISR scaling plot and how this relates to Eq. (12).
[Section 4, Eq. (29)] The fit gives χ^2 ≈ 1 for t_bump(W)=1.355 t_dip(W). Please specify the number of degrees of freedom and the energy range, and clarify whether the uncertainty on T_bd is propagated in this check.
[Section 3, Eq. (15)] The variable y is introduced as y = ln s (or similar) but not defined explicitly. Define y so that d/dy is unambiguous in Eqs. (16)-(18).
[References] Reference [4] is an e-print contribution; if the paper has appeared in a proceedings or journal, update the reference. Reference [15] is a TOTEM paper; the text says 'two different estimates of ρ by TOTEM' but one may come from the TOTEM-D0 combined analysis [6] – clarify.

Circularity Check

1 steps flagged · score 6.0 of 10

The R_bd 'prediction' is a one-parameter fit (c0 fitted), so one headline result reduces partially by construction; T_bd and rho checks retain independent content.

fitted input called prediction [Section 4, Eq. (25) and Fig. 6]
"Hence, we have a one parameter prediction for the ratio Rbd(s)= dσ/dt(t_bump)/dσ/dt(t_dip) = c0 (1+ρ^2(y))/ρ^2(y) , (25) where the constant c0 is given in terms of the function Φ c0= Φ^2(τ_bump)/(τ_dip d/dτ Φ(τ_dip))^2 (26) and we will treat it as a free parameter. In Fig. 6 we plot ratio Rbd (25) for two parametrizations (30) and (31) with c0≃0.03."

The bump-to-dip cross-section ratio R_bd is called a 'one parameter prediction', but the single parameter c0 is not derived or fixed by the model; the text says it will be treated as a free parameter, and the comparison uses c0≃0.03. Since Eq. (25) is proportional to c0, plotting it against data with c0 chosen as 0.03 is a one-parameter fit of the absolute normalization, not a prediction of the ratio's value. The energy dependence through ρ(y) is a genuine consistency check, so the circularity is partial and concerns the normalization of this headline ratio.

full rationale

The central empirical fact, T_bd = 1.355 ± 0.011 from 23 GeV to 13 TeV, is an external data property and is not circular. Its use to infer |t_dip,bump| = τ_dip,bump/R²(s) is a reparameterization until R² is independently identified; the paper does identify R² with σ_tot and verifies matching energy dependences, so this step retains content. The ρ prediction (22) is a self-consistency check: ρ is obtained from derivatives of σ_tot parametrizations fitted to total cross-section data and then compared with measured ρ; it is not circular because the ρ data were not used to fix those derivatives, though it is not a parameter-free first-principles prediction. The clear circular step is R_bd: c0 is left free in Eq. (25) and effectively fitted to the plotted data (c0≃0.03), so the 'computed' bump-to-dip ratio is partly an input. The paper itself also notes that σ_tot² dσ/dt is not universal at the LHC ('the cross-section values can be approximately superimposed by a different function of s'), which limits the central GS claim but is an honest limitation, not a circularity. Eq. (13)'s crossing-even assumption is in tension with the later attribution of the TOTEM ρ decrease to the odderon; this is an internal correctness risk, not a derivation-by-construction. Overall: one headline prediction reduces by construction (partial), while the main empirical regularity and other comparisons are independent; score 6.

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

The model introduces no new particles or forces. It relies on one empirical scale (σ_tot) and an undetermined profile function; the only genuine input beyond proton-scattering data is the analyticity/crossing construction.

free parameters (5)

Interaction radius scale R^2(s) identified with σ_tot(s) = PDG fit: Z=35.45 mb, C=0.308 mb, Y1=42.53 mb, Y2=33.34 mb, eta1=0.458, eta2=0.545; DL fit: A=21.70 mb, B=56.08 mb, alpha
The scaling variable τ=-t R^2(s) uses R^2=σ_tot; σ_tot is parametrized by two literature fits to total cross-section data. The energy slope of R^2 controls the rho prediction.
Power-law exponent β for dip position = β=0.1686 ± 0.0027
Fit to t_dip(W) at LHC energies (Eq. 29); assumed identical to the σ_tot energy dependence.
c0 constant in R_bd prediction = c0 ≈ 0.03
Explicitly treated as a free parameter after Eq. (26) to reproduce R_bd data.
c1 constant in σ_el prediction = not computed
Defined by Eq. (28) in terms of integrals over the unknown shape function Φ; the paper does not provide a value, so the σ_el prediction is not quantitative.
Shape function Φ(τ) = black disk: Φ=2π J_1(√τ)/√τ (illustration only)
Φ is not derived from data; it is either taken from data (Ref [2]) or assumed a specific profile. The R_bd prediction's c0 and the dip/bump positions depend on Φ.

assumptions (5)

domain assumption Geometric scaling: T_el(s,t) depends only on τ = -t R^2(s)
Central ansatz of the paper, Eq. (8); being tested rather than proved.
domain assumption Crossing-even amplitude at high energy: T_el(-s,t) = T*_el(s,t)
Eq. (13); neglects Odderon; contradicted by the paper's own discussion of TOTEM rho.
standard math Analyticity and smoothness of R^2 and Φ allowing first-order Taylor expansion
Eqs. (16)-(17); assumes no nearby singularities; second-order terms claimed small without proof.
standard math Fourier-Bessel transform relation (5) and optical theorem (4)
Standard high-energy scattering formalism.
domain assumption Dip/bump structure corresponds to zero and minimum of Φ
Eq. (21); stated to hold for realistic profiles without proof.

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Cite this review

Pith. "Pith review of Geometric scaling in elastic $pp$ collisions." pith.science (2026). https://pith.science/paper/TJW3N4MI

@misc{pith2026260716182,
  author       = {Pith},
  title        = {Pith review of: Geometric scaling in elastic $pp$ collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJW3N4MI}},
  note         = {Machine review of arXiv:2607.16182}
}

abstract

Geometric scaling was conjectured and observed at the ISR more than 50 years ago. We argue that it still holds at the LHC. We show that the dip-bump structures of the differential elastic cross-sections exhibit a remarkable regularity, namely that the ratio of the bump to dip positions is constant from 23 GeV to 13 TeV. Using crossing symmetry, analicity and the optical theorem, we identify the imaginary and real parts of the scattering amplitude. This allows us to compute the $\rho$ parameter and the ratio of the bump to dip values of the differential $pp$ cross-section. Finally, we discuss the energy dependence of the total elastic cross-section as compared to the total cross-section, and the violation of geometrical scaling outside the dip-bump region at the LHC.

Figures

Figures reproduced from arXiv: 2607.16182 by the authors.

**Figure 1.** Ratio Tbd = |𝑡b|/|𝑡d| for different energies as a function of the data set number from Ref. [2] ( ISR: (1) – 23, (2) – 30, (3) – 45, (4) – 53, (5) – 63 GeV, LHC: (6) – 2.76, (7) – 7, (8) – 8, (9) – 13 TeV). The constancy of Tbd is a strong indication of GS even at the LHC. Let’s explore it assuming for a while that the elastic scattering amplitude 𝑇˜ el(𝑠, 𝑡) is purely imaginary 1 and depends on the scaling variable… view at source ↗

**Figure 2.** Elastic 𝑝 𝑝 cross-section 𝑑𝜎el/𝑑𝑡 [mb/GeV2 ] at the ISR energies in terms of |𝑡| [GeV2 ] – left, and the same scaled according to Eq. (12) – right. We have used, following [8], 𝑅 2 = 𝜎inel rather than 𝑅 2 = 𝜎tot. This does not matter for the ISR since all integrated cross-sections have the same energy dependence (3). 3. Crossing and the real part of the amplitude It is clear that in the vicinity of the dip related t… view at source ↗

**Figure 3.** Left panel: function Φ(𝜏) from Eq. (20) – 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. One can clearly see from [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Fit to dip and bump positions of Eq. (29) with 𝛽 = 0.1686 at LHC energies. In order to compare (22), (25) and (27) with the data we used two analytic parametrizations of the total 𝑝 𝑝 cross-section data. The first one is the COMPETE parametrization [11] quoted in PDG 2…

**Figure 5.** Figure 5: Left: total 𝑝 𝑝 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. Right: Parameter 𝜌 (22) as a function of √ 𝑠 in GeV. Solid magenta line corresp…

**Figure 6.** Figure 6: Ratio Rbd as a function of √ 𝑠 in GeV. Data: brown triangles from Ref. [2], blue circles from Ref. [6]. Theoretical curves (25) as in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Dias De Deus.Geometric Scaling, Multiplicity Distributions and Cross-Sections

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[2] [2]

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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

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