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REVIEW 3 major objections 4 minor 41 references

Extending geometric scaling to the crossing-odd amplitude gives parameter-free rho relations, and adding a maximal-Odderon term accommodates the low 13 TeV proton-proton rho value.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 19:34 UTC pith:7PLIKEJT

load-bearing objection Nice new Ansatz, but the Odderon accommodation claim is off by a sign—the stated parameters increase rho_pp instead of decreasing it. the 3 major comments →

arxiv 2607.16907 v1 pith:7PLIKEJT submitted 2026-07-18 hep-ph

Geometric Scaling and the Odderon

Michal Praszalowicz This is my paper
classification hep-ph PACS 13.85.-t13.85.Lg
keywords geometric scalingOdderonrho parameterelastic proton-proton scatteringcrossing symmetrytotal cross sectionC-odd amplitudeinteraction radius
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that geometric scaling, the idea that elastic scattering depends on energy through a single interaction radius, can be extended to the crossing-odd amplitude, giving it its own radius Q(s). From this, it derives closed formulas for the rho parameter (the ratio of the real to the imaginary forward amplitude) in terms of two interaction radii and their energy derivatives. With a standard fit to total cross sections, these formulas reproduce measured rho values for proton-proton and proton-antiproton scattering up to about 1 TeV, but undershoot the 13 TeV proton-proton point. The paper then adds a small maximal-Odderon term, proportional to ln-squared energy, to the antiproton cross-section and shows it shifts the 13 TeV prediction down to the measured value while altering total cross-sections by only a few millibarns. If correct, this supports the claim that a C-odd exchange, the Odderon, is present at LHC energies.

Core claim

On the paper's own claims, the forward proton-proton and proton-antiproton amplitudes can each be split into crossing-even and crossing-odd pieces that both obey geometric scaling with separate interaction radii R^2 and Q^2. Analytic continuation of the energy variable converts the amplitude's phase into a derivative operator, yielding simple parameter-free relations between the rho parameter, the total cross sections, and the two radii. Fixing the radii from a standard parametrization of the sum and difference of the two total cross sections reproduces the low-energy rho data; only the 13 TeV proton-proton point is missed. Adding the maximal-Odderon function f(A ln^2) to the antiproton tota

What carries the argument

The central object is the geometric-scaling ansatz for the C-odd amplitude, T^-_el(s,t) = -s Q^2(-is) Psi(|t| Q^2(-is)), introduced as the odd counterpart of the even scaling amplitude. The workhorse maneuver is the analytic expansion -is = exp(y - i pi/2), which replaces the radius Q^2(-is) by Q^2(y) - (i pi/2) dQ^2/dy. This splits the amplitude into real and imaginary parts, so the rho parameter and total cross sections become algebraic functions of R^2, Q^2, and dR^2/dy. The paper then exploits a residual gauge freedom in Q^2 to insert an Odderon-shaped function f(y)=A ln^2(s/s3) into the antiproton cross-section while leaving the proton cross-section nearly unchanged.

Load-bearing premise

The whole rho-parameter machinery depends on the untested guess that the crossing-odd amplitude scales geometrically exactly like the even one, T^-_el(s,t) = -s Q^2(-is) Psi(|t| Q^2(-is)); the paper itself labels this a test.

What would settle it

A decisive check is to measure the rho parameter or the total cross-section difference for proton-antiproton scattering at several TeV energies: the maximal-Odderon term predicts rho_pbarp to rise noticeably and sigma_pbarp to exceed sigma_pp by roughly 2-3 mb at LHC energies. If the measured values follow the unmodified parametrization instead, the proposed Odderon explanation fails. A second decisive check is to compare the t-dependence of the C-odd amplitude extracted from the forward relation with future differential cross-section data; if the scaling function Psi does not describe the dip

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • If the ansatz is right, no new model of the even amplitude is needed at 13 TeV; the rho deficit is naturally attributed to a C-odd, Odderon-like contribution.
  • The C-odd scaling function Psi and radius Q^2 make concrete predictions for the difference between pp and pbarp differential cross sections, testable in future runs or at facilities with antiproton beams.
  • The derivative expansion gives an analytic alternative to dispersion relations: future parametrizations of the total cross sections directly determine rho, bypassing numerical dispersion integrals.
  • Once the Odderon parameters are fixed, the same term predicts the size of the pp - pbarp total cross-section difference at LHC energies, turning the rho measurement into a quantitative Odderon probe.
  • The rho values for proton-antiproton scattering above 1 TeV, where no data exist, provide a sharp falsifiable prediction of the modified scenario.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Odderon term is added by hand with parameters chosen rather than fitted; a global fit of the combined parametrization to all forward data would either sharpen the claimed magnitude or expose tensions with existing points.
  • The gauge freedom in the derivation means the same physical shift could be redistributed between proton and antiproton channels; if future data prefer a symmetric adjustment, the qualitative conclusion survives but the specific 2-3 mb prediction would not.
  • One could test assumption (6) directly by checking whether the t-dependence of elastic scattering at ISR energies, where C-odd Reggeon exchanges are sizable, follows the same scaling-law form with a single Q^2.
  • Because the paper only uses forward kinematics, the same C-odd scaling function should show up in the dip and bump structure of differential cross sections; extracting Psi there would provide an independent consistency check.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper extends geometric scaling ideas to the crossing-odd elastic amplitude by postulating T^-_el(s,t) = -s Q^2(-is) Ψ(|t|Q^2(-is)), and derives simple expressions for the total cross-sections and ρ parameters in terms of the interaction radii R^2 and Q^2 (Eqs. (9)-(12)). Using the COMPETE parametrization of σ_tot^{pp,pbarp} as input, the author states that the low-energy ρ data are reproduced. To address the 13 TeV TOTEM/ATLAS ρ_pp deficit, a maximal-Odderon term f(y)=A ln^2(s/s3) is added to σ_tot^{pbarp}; with A=0.08 mb, s3=1 TeV, q=-0.2 mb, the paper claims that ρ_pp moves down to the TOTEM/ATLAS values while total cross-sections are almost unchanged. The core claim is that this modification allows the Odderon interpretation of the 13 TeV data.

Significance. If correct, the paper would provide a simple analytic framework connecting geometric scaling, analyticity, and the Odderon, with explicit formulae that can be checked against data. The derivation of the forward relations is transparent and the algebra is easy to follow. However, the central numerical demonstration is incorrect, and the independent predictive content is weaker than claimed because the COMPETE input is already fitted to ρ data. The geometric-scaling form of the C-odd amplitude is not actually tested by forward data. These issues undermine the main conclusion as it stands.

major comments (3)
  1. [Odderon, Eqs. (18)-(19), Fig. 2] The central numerical claim is contradicted by the paper's own equations. For the stated values (A=0.08 mb, s3=1 TeV, q=-0.2 mb) the extra term in the ρ_pp numerator of Eq. (18) is +(π/8)f'' + f/(2π). At √s=13 TeV, using COMPETE inputs, R^2≈55.12 mb, dR^2/dy≈4.80 mb, Q^2≈-0.199 mb, f≈2.10 mb, f''=0.16 mb. The unmodified ρ_pp is about 0.140, and the added term +0.398 mb raises it to about 0.148, not lowers it. The TOTEM/ATLAS value is ≈0.098, requiring a reduction of ≈0.04. Even with a reversed sign the shift is only -0.007, far too small. Thus the abstract's claim that this modification 'allows to accommodate ρ_pp 13 TeV data' is not supported by the stated parametrization.
  2. [Phenomenology, Eqs. (13)-(15)] The 'reproduction' of the low-energy ρ parameters is largely circular. The COMPETE parametrization (13) was fitted to data that include the ρ measurements, and the integration constant q in Eq. (15) is also adjusted to the same data. Therefore the agreement in Fig. 1 is partly inherited from the input. The text calls Eq. (12) 'parameter free', but q is a free parameter. The authors should clarify what genuinely new predictive content Eq. (12) has, for example by showing a prediction for energies not included in the COMPETE fit, or by testing the relation (12) with an independent σ_tot parametrization.
  3. [Geometric scaling / Phenomenology, Eq. (6)] The forward data used in the paper cannot validate the geometric-scaling Ansatz (6) for the C-odd amplitude. At t=0 only the normalization Ψ(0)=1 enters, so Eqs. (11)-(12) test the analytic behavior of Q^2(y), not the scaling variable |t|Q^2. Therefore the statement 'This result proves that parametrization (6) is a highly probable possibility' is an overreach. A test of the scaling form requires the t-dependence of the C-odd amplitude, which is not analyzed here. The paper should either present such a test or soften the claim to say that the forward data are consistent with the analytic continuation (8) of the C-odd input.
minor comments (4)
  1. [Odderon, Eq. (19)] The parameter s3 is quoted as 1 TeV, but s in the COMPETE parametrization is in GeV^2. Please specify whether s3 means (1 TeV)^2 or 1 TeV in mass units. This ambiguity affects the numerical results and the statement that σ_pbarp changes by '2–3 mb' at LHC energies.
  2. [Odderon, after Eq. (18)] The text says for pbarp above 1 TeV the ρ parameter 'overshoots' the COMPETE parametrization. With the stated positive f and Eq. (18), the ρ_pbarp numerator changes by (π/8)f'' - f/(2π), which is negative for the chosen parameters, so ρ_pbarp is lower, not higher. Please check the figure and the description.
  3. [Eq. (12)] Calling Eq. (12) 'parameter free' is misleading because Q^2 contains the integration constant q from Eq. (15), which is fit to data. The relations are model-dependent through the choice of the COMPETE parameters and q.
  4. [Eq. (16)] The transformation (16) is called a 'gauge' freedom, but it changes the physical ρ parameters and modifies σ_pbarp, so it is not a gauge symmetry in the usual sense. Consider using a different term such as 'reparametrization'.

Circularity Check

2 steps flagged

The 13 TeV rho_pp 'accommodation' is a hand-picked fit, and the COMPETE-based rho 'predictions' reuse data already fitted; no load-bearing self-citation.

specific steps
  1. fitted input called prediction [Phenomenology, Eqs. (12)-(14) and Fig. 1]
    "Now we have all information to compare our predictions with data. In the lower panel we plot rho_pp/pbarp given by Eq. (12). We see that theoretical curves describe data very well."

    The 'predictions' for rho are obtained by inserting the COMPETE parametrization (13) into Eq. (12). The COMPETE parameters were fitted to total cross-section and rho data. Hence the agreement between Eq. (12) and the rho data is not an independent test: the input parametrization already contains the information about rho, so the output is statistically constrained by the same data. The paper's conclusion that the C-odd ansatz (6) is 'a highly probable possibility' therefore goes beyond what the comparison can establish.

  2. fitted input called prediction [Odderon section, Eqs. (18)-(19), Fig. 2]
    "As an example, let's choose A = 0.08 mb, s3 = 1 TeV and q = -0.2 mb. The results for the rho parameters are shown in Fig. 2 as red dashed lines. We see that with this choice we are able to touch TOTEM data without spoiling rho_pp below 1 TeV."

    The values A, s3, q are free constants selected in the paper; they enter directly in rho_pp through Eq. (18). Presenting the outcome that the 13 TeV TOTEM/ATLAS points are 'touched' as a success is equivalent to saying the model fits the data after choosing its parameters for that purpose. The paper even admits that a true refit is 'beyond the scope' and that it instead 'choose[s] some reasonable values to see if adding the Odderon in this way is at all possible.' Therefore the claimed accommodation is a manual fit, not a prediction, and the conclusion that the Odderon can accommodate the data reduces to the hand-picked parameter set.

full rationale

The derivation of Eqs. (9)-(12) from the geometric-scaling Ansatze (5)-(6) is a legitimate, self-contained mathematical step; no circularity appears in that part. The problems arise at the comparison stage. First, Eq. (12) is called 'predictions' but is evaluated with the COMPETE parametrization (13), whose parameters were fitted to the very rho data being 'reproduced'; the agreement is therefore partly an artifact of the shared fit. Second, the Odderon modification (18)-(19) is tested by choosing A, s3, q so as to touch the 13 TeV TOTEM points, and the resulting agreement is then reported as 'we are able to touch TOTEM data'; this is a fit presented as an accommodation, not an independent derivation. No load-bearing self-citation is present. The numerical sign of the f contribution appears to increase rho_pp at 13 TeV rather than decrease it, which is an independent correctness concern but does not alter the circularity assessment.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claim rests on the new C-odd geometric-scaling Ansatz and on the COMPETE total-cross-section fit. The visible free parameters are q, A, s3; the COMPETE constants are inherited fitted values. No new particles or entities are introduced; the Odderon is taken from the literature.

free parameters (3)
  • COMPETE total cross-section parameters (Z, C, Y1, Y2, η1, η2, s0, s1) = Z=35.45 mb, C=0.308 mb, Y1=42.53 mb, Y2=33.34 mb, s0=28.94 GeV^2, s1=1 GeV^2, η1=0.458, η2=0.545
    Inherited from the COMPETE fit [3]; the rho 'predictions' in Eq. (12) are computed from these fitted values.
  • Integration constant q for Q^2(y) = 0 mb (Fig. 1), -0.2 mb (Fig. 2)
    Q^2 is determined up to a constant by Eq. (15); q is chosen by hand, not derived.
  • Maximal Odderon parameters A and s3 = A=0.08 mb, s3=1 TeV
    Hand-picked 'reasonable values' to make rho_pp touch TOTEM 13 TeV data; no fit or uncertainty.
axioms (5)
  • domain assumption Elastic pp amplitude satisfies geometric scaling: T_el^pp(s,t) = i s R^2(s) Φ(|t|R^2(s)) (Eq. 1).
    Motivated by empirical scaling of bump/dip positions and cross-section ratios; assumed as the starting point.
  • domain assumption Crossing-even amplitude is obtained by R^2(-is) analytic continuation: T^+ = i s R^2(-is) Φ(|t|R^2(-is)) (Eq. 5).
    Based on Ref. [31] trick; not derived from first principles.
  • ad hoc to paper Crossing-odd amplitude has the analogous geometric-scaling form T^- = -s Q^2(-is) Ψ(|t|Q^2(-is)) (Eq. 6).
    New Ansatz, stated as a test; the entire derivation of rho_pp and rho_pbar-p depends on it.
  • standard math First-order Taylor expansion R^2(-is) ≈ R^2(y) - i(π/2)dR^2/dy is valid; higher-order terms negligible.
    The paper asserts higher-order corrections can be safely neglected, checked only a posteriori by data agreement.
  • domain assumption COMPETE parametrization (13) describes the total pp and pbar-p cross-sections over the energy range.
    Taken from the prior global fit; the paper does not refit and relies on it for both sigma and hence rho.

pith-pipeline@v1.3.0-alltime-deepseek · 8222 in / 16562 out tokens · 160601 ms · 2026-08-01T19:34:06.967281+00:00 · methodology

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Cite this review

Pith. "Pith review of Geometric Scaling and the Odderon." pith.science (2026). https://pith.science/paper/7PLIKEJT

@misc{pith2026260716907,
  author       = {Pith},
  title        = {Pith review of: Geometric Scaling and the Odderon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7PLIKEJT}},
  note         = {Machine review of arXiv:2607.16907}
}
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read the original abstract

Based on geometric scaling of elastic $pp$ and possibly $p\bar{p}$ cross-sections, we derive simple formulae for the complex crossing-even and crossing-odd scattering amplitudes in terms of two interaction radii: $R(s)$ and $Q(s)$, respectively. Employing the COMPETE parametrization of the total cross-sections, we reproduce $\rho^{pp}$ and $\rho^{p\bar{p}}$ parameters with high accuracy, however we miss the 13~TeV TOTEM and ATLAS points. We show that a slight modification of the $p\bar{p}$ total cross-section corresponding to the Odderon, allows to accommodate $\rho^{pp}$ 13~TeV data.

Figures

Figures reproduced from arXiv: 2607.16907 by Michal Praszalowicz.

Figure 1
Figure 1. Figure 1: FIG. 1. Upper panel – total cross-sections, lower panel – [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Total [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

discussion (0)

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Reference graph

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