Odderon in the light of collider low-$t$ data

E.G.S. Luna; M.G. Ryskin; V.A. Khoze

REVIEW 4 major objections 4 minor 12 references

Odderon in the light of collider low-$t$ data

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Including a small Odderon exchange improves the fit to low-|t| elastic data, but its imprint on rho at 13 TeV is at most 0.004, ten times smaller than TOTEM's value.

desk verdict Serious and honest fit, but the Odderon sign and size are conditional on an untested t=0 form factor that the authors themselves concede in footnote 2. read the letter →

arxiv 2411.15275 v1 pith:JMTAF6HJ submitted 2024-11-22 hep-ph hep-ex

E.G.S. Luna , M.G. Ryskin , V.A. Khoze This is my paper · ORCID

classification hep-phhep-ex

keywords OdderonelasticscatteringCoulomb-nuclearinterferencetwo-channeleikonalmodelrhoparameterPomeronLHClowmomentumtransfer

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 tries to establish how large the Odderon—the charge-odd exchange that does not fall, or falls very slowly, with energy—can be in high-energy elastic proton scattering. The authors fit $pp$ and $\bar p p$ data in the Coulomb-nuclear interference region, $|t| < 0.1$ GeV$^2$, over $\sqrt{s} = 50$ GeV to 13 TeV, using a two-channel eikonal model that screens the Odderon by the Pomeron. Adding an Odderon with intercept $\alpha_O(t) = 1$ lowers $\chi^2$ from 726 to 560 for 504 degrees of freedom. The physical $C$-odd asymmetry in the forward real-to-imaginary ratio is $\delta\rho = (\rho_{\bar p p} - \rho_{pp})/2 \le 0.004$, an order of magnitude smaller than the value behind TOTEM's Odderon claim, and the needed sign is opposite to the perturbative QCD three-gluon exchange prediction. If correct, the Odderon exists at the LHC but is far weaker, and with a sign puzzle, than previously claimed.

What carries the argument

The carrying object is the two-channel eikonal amplitude (Eq. 2), built from an opacity $\Omega(s,b) = \Omega_P(s,b) + \Omega_O(s,b)$. Each bare exchange is Fourier-Bessel transformed to impact parameter, and unitarization mixes two diffractive eigenstates through the coupling $\gamma$; this is the mechanism that screens the seed Odderon by the C-even Pomeron. The Odderon input is $\beta_O^2(t)\, \eta_O(t)\, (s/s_0)^{\alpha_O(t)}$ with the odd signature factor $\eta_O(t) = -i e^{-i\pi\alpha_O(t)/2}$ and the trajectory fixed to the maximal QCD value $\alpha_O(t) = 1$. The relative signs of the three unitarization terms in Eq. 2 control whether the physical $C$-odd contribution is suppressed and which sign it takes.

What would settle it

Measure $\rho$ for $pp$ and $\bar p p$ at the same high energy with a total uncertainty below 0.002; if $|\rho_{\bar p p} - \rho_{pp}|/2$ exceeds 0.004 at 13 TeV, the paper's central bound is falsified. A simpler check is to redo the fit with a single-channel eikonal or an alternative screening scheme and see whether the extracted $\delta\rho$ remains below 0.004.

Watch

Extended reading notes

Core claim

The authors show that the global low-$|t|$ elastic dataset, including both TOTEM and ATLAS/ALFA measurements, is described better when a $C$-odd Odderon exchange with $\alpha_O(t)=1$ is included in the two-channel eikonal amplitude: $\chi^2 = 560$ for 504 degrees of freedom with the Odderon versus 726 without it. The best-fit Odderon-proton coupling is $\beta_O(0) = 0.90 \pm 0.18$, smaller than the Pomeron coupling, and after eikonal screening the final $C$-odd contribution to $\rho$ at 13 TeV is $\delta\rho = (\rho_{\bar p p} - \rho_{pp})/2 \le 0.004$, not the $\delta\rho \approx 0.04$ claimed by TOTEM. The required sign of the Odderon amplitude is opposite to that of the pQCD three-gluon exchange, and the paper suggests this can be reconciled if $\beta_O(t)$ vanishes or strongly decreases at $t = 0$, in which case the dominant $C$-odd contribution at $t = 0$ comes from the Pomeron-Odderon cut with the opposite sign.

Load-bearing premise

The bound depends on the specific two-channel eikonal screening formula and on the assumption that the Odderon grows with energy at the maximal QCD rate; change either, and the extracted Odderon coupling and the 0.004 bound move.

Editorial extensions

If this is right

The Odderon is not ruled out by low-$|t|$ data, but its physical imprint on the forward $\rho$ parameter at 13 TeV is at most 0.004, an order of magnitude below the $\delta\rho = 0.04$ used in TOTEM's original Odderon claim.
The sign of the required $C$-odd amplitude is opposite to the bare perturbative three-gluon exchange, so the $t = 0$ $C$-odd effect would have to come from the Pomeron-Odderon cut if $\beta_O(0)$ vanishes.
Including the Odderon improves the global fit from $\chi^2 = 726$ to 560 for the same data, so future global analyses of elastic scattering should include a $C$-odd term even if its coupling is small.
The TOTEM normalizations come out well above unity, up to 1.15 at 13 TeV, so the absolute normalization of TOTEM data is a major driver of the extracted cross sections and $\rho$ values.

Reading between the lines

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

By extension, if the ATLAS/ALFA normalization is correct, the TOTEM normalizations needed here (1.077 to 1.15) suggest the apparent Odderon signal may be partly a normalization artefact; a single high-precision $\rho$ measurement at 13 TeV with unified normalization would settle which dataset is right.
The same two-channel screening logic, applied to the diffractive dip region, predicts a specific $C$-odd asymmetry in $d\sigma/dt$ between $pp$ and $\bar p p$; measuring that asymmetry at the LHC or in a future collider would test whether the sign and size extracted here are correct.
A QCD determination of the Odderon intercept away from 1 would change the bound: a lower intercept makes the Odderon fade faster and pushes $\delta\rho$ lower, while a higher one would revive a TOTEM-sized effect.

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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

Both the reader's report and the stress-test note land on the right spot. The paper's central claim—that a small Odderon with sign opposite to pQCD improves the low-|t| fit—is really a conditional statement, and the authors flag the condition in footnote 2. If βO(t) develops a node at t=0, the dominant C-odd contribution at t=0 comes from the Pomeron–Odderon cut and flips sign. Since the fitted data all sit at finite |t|, the t=0 extrapolation is doing all the work for δρ and for the sign. That is a load-bearing caveat, not a nuisance detail.

What is genuinely new is applying the two-channel eikonal model to extract an Odderon term from the CNI region at collider energies, and the reported χ² improvement (726→560) is large. The paper also does something useful by treating TOTEM normalization factors as free parameters; the result that ATLAS normalizations sit near 1 while TOTEM needs N13=1.15 is worth knowing, even if the size of that rescaling invites caution. The comparison to the pQCD three-gluon sign is a legitimate external benchmark, but as the reader notes, the magnitude and sign are outputs of the same fit, not independent predictions.

The soft spots are mostly concentrated in what is not shown. No significance test is given for the added Odderon parameters, δρ is quoted as an upper limit with no uncertainty, and the trajectory is fixed at αO=1 with only the slope D varied (0.1A–0.9A). The D-independence of σtot and ρ is reassuring, but it does not address the node-in-βO scenario that footnote 2 concedes would flip the sign. That makes the headline conclusion—'sign opposite to pQCD, δρ≤0.004'—a property of the chosen exponential form, not a robust extraction.

Despite these caveats, the paper is coherent and unusually honest for a conference proceeding: footnote 2 is exactly where too many authors would have buried the problem. The fit is plausible and the data handling is transparent. It deserves a serious referee—one who should ask for a node-βO fit, a proper likelihood treatment of the extra parameters, and uncertainties on δρ. I would bring it to a reading group if we are discussing Odderon phenomenology, and I would cite it once those caveats are addressed.

Send it to peer review, with a recommendation for major revision rather than acceptance as is.

Referee Report

4 major / 4 minor

Summary. The paper analyzes proton-proton and proton-antiproton elastic scattering data in the Coulomb-nuclear interference region (|t| < 0.1 GeV^2, sqrt(s) = 50 GeV to 13 TeV) using a two-channel eikonal model that adds a C-odd Odderon exchange to the dominant C-even Pomeron. The authors report that including the Odderon improves the global chi^2 from 726 to 560 for 504 degrees of freedom, yielding a best-fit Odderon coupling beta_O(0) = 0.90 ± 0.18, with a slope D = A/2 fixed to half the Pomeron vertex slope. The central conclusion is that the required C-odd contribution to the forward real-to-imaginary ratio at 13 TeV is small, delta_rho <= 0.004, i.e. about ten times smaller than the value originally claimed by TOTEM, and that the sign of the extracted Odderon amplitude is opposite to the perturbative QCD three-gluon exchange prediction. The paper also finds large normalization rescaling factors for TOTEM data (up to N13 = 1.15).

Significance. If the result is robust, it would significantly constrain the size and sign of a possible high-energy Odderon contribution, and would provide a concrete quantitative bound that can be compared with model predictions. The paper's use of a two-channel eikonal to account for screening of the Odderon by Pomeron exchanges is a valuable feature, and the global fit includes both TOTEM and ATLAS/ALFA data with explicit normalization penalties. However, the strength of the claim is limited by the absence of a statistical significance test for the chi^2 improvement, by the very large normalization rescaling of TOTEM data, and by the fact that the sign and smallness of delta_rho depend on an untested assumption about the t-dependence of the Odderon-proton vertex.

major comments (4)

[Section III, Eq. (1), Table I] The paper reports a chi^2 improvement from 726 to 560 when the Odderon is introduced, but no significance test (e.g., a likelihood-ratio or F-test accounting for the added parameters beta_O(0) and D, together with the free TOTEM normalization factors) is provided. Since the normalization factors themselves can absorb systematic offsets, the reader cannot judge whether the improvement is statistically meaningful. This is load-bearing for the central claim that the Odderon term is required by the data.
[Section III, Table I and Fig. 1] The TOTEM normalization factors N7 = 1.077, N8 = 1.121, and N13 = 1.15 rescale the TOTEM data by up to 15%, which is far larger than the typical normalization uncertainties quoted by the experiments. With the 13 TeV TOTEM data multiplied by 1.15, the chi^2 improvement may be driven more by rescaling than by a physical Odderon contribution. The authors should demonstrate that the conclusion survives when the normalizations are constrained to their nominal values (or when the penalty terms are weighted more strongly), and should discuss the compatibility of N13 = 1.15 with the published TOTEM luminosity and normalization uncertainties.
[Section IV and footnote 2] The sign of the Odderon contribution and the bound delta_rho <= 0.004 rely on the assumed form beta_O(t) = beta_O(0) e^{Dt/2}, which is nonzero at t = 0. The footnote concedes that if beta_O(t) vanishes or strongly decreases at t = 0, the dominant C-odd contribution at t = 0 comes from the Pomeron-Odderon cut and 'has the opposite sign.' Since all fitted data are at finite |t|, the t = 0 extrapolation is not tested by the fit. The paper should present fits with a node in beta_O(t) (e.g., beta_O(t) proportional to t or another form vanishing at t = 0) and report whether the sign and size of the extracted delta_rho are stable; as written, the quoted sign is a property of the chosen exponential vertex, not a model-independent extraction.
[Section II and Section III] The Odderon trajectory is fixed to alpha_O(t) = 1 and the screening is fixed to the specific two-channel eikonal form of Eq. (2). The paper tests only the dependence on the slope D, varying it from 0.1A to 0.9A, and does not explore alternative unitarization schemes or variations of alpha_O(0) and alpha'_O. The extracted beta_O(0) and the bound delta_rho <= 0.004 are therefore conditional on these model choices. The authors should either justify these choices in more detail or scan over a plausible range of alpha_O(0) and alpha'_O to quantify the resulting uncertainty on delta_rho.

minor comments (4)

[Section IV] The bound delta_rho <= 0.004 is quoted without any uncertainty or confidence level; since beta_O(0) has a reported error, the final C-odd contribution should include a propagated uncertainty.
[Section II, Eq. (7)] There is a typographical error in the text following Eq. (7): 'beta_O(t)) = beta_O(0)e^{Dt/2}' contains a stray parenthesis; this should read 'beta_O(t) = beta_O(0)e^{Dt/2}'.
[Section I] The paper states that ATLAS/ALFA 'confirmed' the TOTEM value of rho, but the ATLAS result is based on a different total cross-section value that is about 5% lower; it would be useful to state explicitly how the rho values compare after accounting for this difference.
[Section III, Fig. 1] The right panel of Fig. 1 shows the t-dependence at 13 TeV, but the vertical axis label is not visible in the reproduced figure; please ensure that the axis is labeled with the differential cross-section and its units.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Odderon parameters are fit outputs, and the paper flags the sign's dependence on the βO(t) ansatz.

full rationale

This paper is a fit-based extraction, not a derivation that equates a result to its own input. The Odderon coupling βO(0), the slope D, and the normalization factors are free parameters determined by minimizing χ2 against the low-|t| elastic dσ/dt data. The reported quantities — the improved χ2, the sign of the fitted Odderon coupling, and the small forward C-odd contribution δρ ≤ 0.004 — are all consequences of that fit, not independent predictions. No fitted parameter is relabeled as a prediction, and no equation defines one result in terms of the conclusion it is meant to establish. The paper's own footnote 2 explicitly concedes that the extracted sign depends on the assumed form βO(t) = βO(0)e^{Dt/2}, noting that if βO(t) has a node at t = 0 the dominant forward C-odd contribution would come from the Pomeron–Odderon cut and would have the opposite sign. This is a model-dependence caveat, not circular reasoning. The self-citations to [4] and [12] are not load-bearing: [4] is a prior application of the same eikonal model, and [12] is an independent pQCD three-gluon-exchange calculation; the present paper states its own equations, performs its own fit, and does not invoke those citations as the justification for its central claim. Therefore no circular step is present, and the appropriate score is 0.

Assumptions & free parameters 11 free parameters · 6 assumptions · 0 invented entities

The central result is an extraction from a global fit. The quantitative bound on delta_rho is controlled by model assumptions (eikonal scheme, alpha_O=1) and by normalization rescaling of TOTEM data; the paper varies only the Odderon slope D.

free parameters (11)

beta_P(0) = 2.259 +/- 0.016
Pomeron-proton coupling, fitted to the low-t data ensemble.
epsilon = 0.1180 +/- 0.0020
Pomeron trajectory intercept shift, fitted.
alpha'_P = 0.128 +/- 0.022 GeV^-2
Pomeron trajectory slope, fitted.
A = 4.78 +/- 0.21 GeV^-2
Exponent coefficient in Pomeron vertex beta_P(t), fitted.
B = 6.7 +/- 1.1 GeV^-4
Second-order coefficient in beta_P(t), fitted.
C = 17.7 +/- 4.0 GeV^-6
Third-order coefficient in beta_P(t), fitted.
beta_O(0) = 0.90 +/- 0.18
Odderon-proton coupling, fitted; its sign determines the C-odd contribution.
D = A/2 (fixed; varied from 0.1A to 0.9A)
Slope of Odderon vertex; fixed to A/2, checked to not affect sigma_tot or rho.
Normalization factors N7, N8, N13 (ATLAS) = 1.015, 1.003, 1.009
Free parameters in chi^2 to absorb ATLAS dataset normalization.
Normalization factors N7, N8, N13 (TOTEM) = 1.077, 1.121, 1.15
Free parameters in chi^2 to absorb TOTEM dataset normalization; values are large relative to typical quoted uncertainties.
gamma
Parameter in the two-channel eikonal formula (Eq. 2) controlling the mixing of diffractive eigenstates; not reported in the text.

assumptions (6)

domain assumption Regge pole form for Pomeron and Odderon amplitudes (Eqs. 4 and 7)
The scattering amplitudes are assumed to be dominated by Regge poles with simple exponential vertices and linear trajectories.
domain assumption Two-channel eikonal unitarization (Eq. 2)
The data are described by an eikonal model with three Good-Walker-like components; this is a standard but non-unique unitarization scheme.
ad hoc to paper Odderon trajectory alpha_O(t)=1
Fixed to the largest QCD value (three-gluon exchange) rather than fitted; the conclusion about sign depends on this choice.
standard math Pion loop insertion in Pomeron trajectory h(pi pi) from [6]
Taken from cited literature; shape of the trajectory at low t.
domain assumption Normalization factors absorb all TOTEM-ATLAS discrepancies
The chi^2 procedure assumes the differences between the two experiments are purely multiplicative normalization offsets.
standard math CNI formula with Bethe phase (Eq. 8)
Standard Coulomb-nuclear interference formula used to compute dsigma/dt.

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Cite this review

Pith. "Pith review of Odderon in the light of collider low-$t$ data." pith.science (2026). https://pith.science/paper/JMTAF6HJ

@misc{pith2026241115275,
  author       = {Pith},
  title        = {Pith review of: Odderon in the light of collider low-$t$ data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JMTAF6HJ}},
  note         = {Machine review of arXiv:2411.15275}
}

abstract

Odderon is the $C$-odd amplitude that does not fall (or decrease very slowly) with energy. The expected amplitude is small and mainly real. Therefore, extracting it from the data on top of a much larger $C$-even contribution is challenging. The only chance is to consider the very low $|t|$ region of Coulomb nuclear interference or the diffractive dip region. Here we perform the analysis of elastic scattering $pp$ and $\bar pp$ data at low momentum transfer $|t| < 0.1$ GeV$^2$ within large collider energy interval $\sqrt s = 50$ GeV $-$ 13 TeV in order to evaluate quantitatively the possible Odderon contribution. We use the two-channel eikonal model, which naturally accounts for the screening of the Odderon amplitude by the $C$-even (Pomeron) exchanges.

Figures

Figures reproduced from arXiv: 2411.15275 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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