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arxiv: 2605.25636 · v1 · pith:6U7JEO5Wnew · submitted 2026-05-25 · ✦ hep-ph · hep-th· nucl-th

Odderon Form Factors in Reggeized Spin-2 Pomeron and Spin-3 Odderon Exchange in pp and pbar p Elastic Scattering

Dominador F. Vaso , Jr. , Prin Sawasdipol , Jingle B. Magallanes , Chakrit Pongkitivanichkul , Daris Samart This is my paper

Pith reviewed 2026-06-29 21:47 UTC · model grok-4.3

classification ✦ hep-ph hep-thnucl-th
keywords odderonpomeronelastic scatteringform factorsregge theorypp scatteringpbarp scatteringhigh energy
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The pith

An exponential Odderon-proton form factor fits high-energy elastic scattering data far better than dipole, polynomial or Gaussian alternatives.

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

The paper tests seven different parametrizations of the Odderon-proton vertex inside a Reggeized spin-3 Odderon plus spin-2 Pomeron model for pp and pbarp elastic scattering. Six standard forms produce comparable but mediocre fits to the combined TOTEM and Tevatron data sets, while the one-parameter exponential form yields a distinctly lower chi-squared. The improvement is localized to the vertex because the fitted Regge slopes and couplings stay stable across choices. The exponential choice corresponds to a Gaussian transverse profile whose radius is hadronic in size, offering a direct link between the pp versus pbarp dip-bump difference and the spatial structure of C-odd exchange.

Core claim

Among seven Odderon-proton form-factor parametrizations tested in a global fit to TOTEM pp data at 2.76-13 TeV and Tevatron pbarp data at 1.8-1.96 TeV, the exponential form F_O(t) = exp[-B|t|/2] gives chi^2_red = 0.98 for 138 degrees of freedom while the other six forms (dipole, polynomial, Gaussian, hybrid) give 1.44-1.48; the extracted couplings and trajectory slopes remain stable, showing that the fit improvement arises from the vertex itself rather than compensatory changes in the Regge dynamics.

What carries the argument

Covariant spin-3 projector for the Reggeized Odderon kept factorized from the scalar trajectory kernel, combined with different Odderon-proton form-factor parametrizations.

If this is right

  • The Odderon-proton interaction has an effective transverse radius of hadronic size corresponding to a Gaussian impact-parameter profile.
  • The t-range where single-Regge exchange works shrinks with rising energy, signalling the onset of absorptive and unitarity corrections.
  • The dip-bump structure difference between pp and pbarp scattering is directly tied to the transverse profile of the C-odd color-singlet exchange.
  • Fitted Regge couplings and slopes are robust against the choice among the tested form factors.

Where Pith is reading between the lines

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

  • Models of soft diffraction that treat the Odderon vertex as more peripheral than the Pomeron vertex may be preferred by data.
  • Extending the same exponential form factor to higher energies could predict the rate at which the single-exchange window closes.
  • Direct comparison of the extracted radius with lattice or holographic estimates of the Odderon size would test the peripheral-interaction picture.

Load-bearing premise

The single-Regge-exchange description with factorized spin projectors remains accurate throughout the fitted t-range at each energy.

What would settle it

A precise measurement of the differential cross section at energies above 13 TeV or at |t| values beyond the fitted range that deviates from the single-exchange prediction while the exponential form factor is held fixed would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.25636 by Chakrit Pongkitivanichkul, Daris Samart, Dominador F. Vaso, Jingle B. Magallanes, Jr., Prin Sawasdipol.

Figure 1
Figure 1. Figure 1: FIG. 1: Elastic scattering via Pomeron and Odderon exchange: (a) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The differential cross section of elastic [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The differential cross section for elastic [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Normalized impact-parameter profiles [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Bootstrap/Monte Carlo correlation heatmap for the FF7 global-fit parameters. Strong positive correlations [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
read the original abstract

We investigate the form-factor dependence of Reggeized tensor Pomeron and Odderon exchanges in high-energy elastic $pp$ and $p\bar p$ scattering. The spin structure is implemented through explicit covariant spin-2 and spin-3 projectors, kept factorized from the Reggeized scalar kernels, so that vertex effects can be separated from trajectory dynamics. Seven Odderon--proton form-factor parametrizations are tested against a global dataset including TOTEM $pp$ data at $\sqrt{s}=2.76$, $7$, $8$, and $13$~TeV and Tevatron $p\bar p$ data at $\sqrt{s}=1.80$ and $1.96$~TeV. A clear hierarchy is found. Six dipole, polynomial, Gaussian, and hybrid parametrizations give comparable fit qualities, $\chi^2_{\rm red}\simeq 1.44$--$1.48$, whereas a one-parameter exponential form, $F_{\mathbb O}(t)=\exp[-B|t|/2]$, yields $\chi^2_{\rm red}=0.98$ for 138 degrees of freedom. The fitted couplings and Regge slopes remain comparatively stable across the form-factor choices, indicating that the improvement is driven mainly by the Odderon--proton vertex rather than by large compensating shifts in trajectory parameters. The exponential form admits an impact-parameter interpretation as a Gaussian transverse profile, with an effective radius $\sqrt{\langle b^2\rangle}=\sqrt{2B}\,\hbar c$. The extracted radii are of hadronic size and suggest a peripheral soft Odderon interaction. The shrinking $t$-range over which the single-Regge-exchange description remains accurate at increasing energy indicates the onset of absorptive and unitarity corrections. These results provide a compact phenomenological framework for connecting the $pp/p\bar p$ dip--bump difference with the transverse structure of $C$-odd color-singlet exchange.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript examines the dependence of Reggeized spin-2 Pomeron and spin-3 Odderon exchanges on the choice of Odderon-proton form factor in high-energy elastic pp and p¯p scattering. Using covariant spin projectors factorized from Reggeized kernels, seven form-factor parametrizations are tested in a global fit to TOTEM and Tevatron data. The authors report that an exponential form factor yields a substantially better reduced chi-squared (0.98 for 138 dof) than six other dipole, polynomial, Gaussian, and hybrid forms (χ²_red ≈ 1.44–1.48), with fitted couplings and Regge slopes remaining stable across choices.

Significance. If the single-Regge-exchange framework remains valid over the fitted kinematic range, the result indicates that the exponential Odderon form factor, interpretable as a Gaussian impact-parameter profile with hadronic radius, provides a superior description of the C-odd exchange. The explicit reporting of χ²_red values and parameter stability across models is a positive feature, offering a concrete phenomenological benchmark for future studies of the Odderon.

major comments (1)
  1. [Abstract] Abstract: the abstract states that 'the shrinking t-range over which the single-Regge-exchange description remains accurate at increasing energy indicates the onset of absorptive and unitarity corrections.' The global fit nevertheless incorporates data at √s = 2.76–13 TeV without apparent restriction to energy-dependent validity windows. If points outside these windows are included, the one-parameter exponential's improvement (χ²_red = 0.98) may partly compensate for missing higher-order effects rather than isolate a genuine vertex structure, weakening the claim that the hierarchy is 'driven mainly by the Odderon-proton vertex'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive comment on the abstract. We address the point below and indicate the revision we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the abstract states that 'the shrinking t-range over which the single-Regge-exchange description remains accurate at increasing energy indicates the onset of absorptive and unitarity corrections.' The global fit nevertheless incorporates data at √s = 2.76–13 TeV without apparent restriction to energy-dependent validity windows. If points outside these windows are included, the one-parameter exponential's improvement (χ²_red = 0.98) may partly compensate for missing higher-order effects rather than isolate a genuine vertex structure, weakening the claim that the hierarchy is 'driven mainly by the Odderon-proton vertex'.

    Authors: The global fit is performed uniformly over the full dataset without energy-dependent t-cuts, as the single-Regge-exchange framework is applied consistently to isolate the effect of the Odderon-proton form factor. The abstract statement on the shrinking t-range reflects the empirical observation that the model's applicability narrows with rising energy, signaling the eventual importance of absorptive corrections. The exponential form factor's markedly lower χ²_red (0.98 vs. 1.44–1.48) occurs together with stable fitted values for the Regge slopes and couplings across all seven parametrizations; such stability would be unlikely if the improvement were mainly from compensating missing higher-order terms. We therefore maintain that the hierarchy is driven primarily by the vertex. To address the concern explicitly, we will add a clarifying paragraph in the results section stating that the form-factor comparison is conducted inside the single-exchange approximation and that the observed hierarchy persists despite the model's known limitations at the highest energies. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in form-factor comparison

full rationale

The paper conducts an empirical comparison of seven Odderon-proton form-factor parametrizations by fitting each to the same global dataset of TOTEM and Tevatron elastic scattering data, then reports the resulting reduced chi-squared values (with the exponential form yielding 0.98 for 138 dof versus 1.44-1.48 for the others). This constitutes a standard model-selection exercise within a phenomenological Regge-exchange framework rather than any derivation in which a claimed result reduces by construction to its own inputs. No equations are presented that equate a fitted quantity to a prediction, no self-citations serve as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The stability of extracted couplings and slopes is reported as an observation from the fits themselves. The analysis is therefore self-contained against external benchmarks with no circular steps.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on multiple fitted parameters (B, couplings, Regge slopes) and standard Regge-theory assumptions; no new entities are introduced.

free parameters (3)
  • B (exponential slope)
    Single free parameter of the winning form factor, fitted to data
  • Odderon and Pomeron couplings
    Fitted to the global dataset and reported as stable
  • Regge slopes alpha'
    Fitted trajectory parameters that remain stable across form-factor choices
axioms (2)
  • domain assumption Factorization of covariant spin-2 and spin-3 projectors from the Reggeized scalar kernels
    Explicitly stated as the method used to separate vertex effects from trajectory dynamics
  • domain assumption Single-Regge-exchange approximation is valid in the t-range used for the fits
    Used to extract chi2 values; the abstract notes the range shrinks with energy

pith-pipeline@v0.9.1-grok · 5942 in / 1589 out tokens · 42192 ms · 2026-06-29T21:47:05.040489+00:00 · methodology

discussion (0)

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Reference graph

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