Across the tested fits, a stable, nonzero C-odd contribution was not required to describe elastic proton-proton and proton-antiproton scattering in the diffractive dip region. Extra freedom instead left the odd contribution compatible with zero or weakly constrained, and a second Odderon-like term was not resolved independently.
The study examined possible C-odd effects in elastic proton-proton and proton-antiproton scattering in the diffractive dip region. At higher |t|, the model diagnostic showed that increasing the C-odd term would move the two cross sections in opposite directions.
What the fits tested
The analysis used 452 published collider measurements of elastic differential cross sections. The sample covered antiproton-proton data at 546 GeV, 1.8 TeV and 1.96 TeV, alongside proton-proton data at 2.76, 7, 8 and 13 TeV.
The model was applied to collider data above 500 GeV and over a momentum-transfer interval of 0.2 < |t| < 1 GeV². It represented scattering with effective C-even and C-odd amplitudes; their relative phases were fixed by signature factors, and their real and imaginary parts were linked by crossing.
To test stability, the analysis used a sequence of model variants. A1 fixed the Odderon-like trajectory at αO(t)=1; A2 freed the odd-residue slope; A3 also released the odd intercept and trajectory slope. A4 added a fourth Pomeron-like term, while A5 added a second Odderon-like term.
Each variant was fitted by global least squares with MIGRAD, and local parameter errors were based on the chi-squared hypersurface around the minimum. The analysis compared two error treatments: the published experimental errors and a diagnostic 13 TeV ensemble in which the same points had their 13 TeV systematic errors multiplied by a factor before being added in quadrature with statistical errors.
The odd contribution stayed small or uncertain
In A1, the Odderon-like coupling was compatible with zero. When the odd-residue slope was fitted independently in A2, the coupling was nonzero but the slope was large—about 8 to 9 GeV−2. At |t| near 0.5 GeV², the residue factor was only exp(−4) to exp(−4.5), and the fitted term was small in the dip region.
Releasing the odd intercept and trajectory slope in A3 did not yield a tightly determined odd sector. The intercept remained compatible with zero, and the odd-residue slope was compatible with zero but carried large uncertainty. The C-odd real part was no more than about 3% to 4% of the effective C-even real part in the dip region.
The sequence as a whole left the odd contribution compatible with zero or weakly constrained. The second Odderon-like term was not resolved as an independent contribution, so the tested variants did not show a stable nonzero C-odd term.
Extra fit quality came mainly from the even sector
The fourth-term variant A4 reported a reduced chi-squared of 1.79, compared with 1.86 for A3. The second-Odderon variant A5 reported 1.84. After the 13 TeV reweighting, the comparable values were 1.50 for B3, 1.44 for B4 and 1.51 for B5. The reported improvement was mainly in the effective C-even amplitude.
Reweighting mainly reduced the 13 TeV partial contribution. It did not remove larger contributions from the 7 TeV, 2.76 TeV or 1.96 TeV samples. The fit described the 8 TeV data better than the 7 TeV data, a pattern the analysis said pointed to tension between datasets.
At higher |t|, the fit tended to undershoot both the 2.76 TeV proton-proton and 1.96 TeV antiproton-proton data. The study judged the missing contribution mainly C-even; increasing the C-odd term would move the two cross sections in opposite directions.
A result about this model, not a final verdict
Those conclusions apply to the effective parametrisation tested here. With one or two Odderon-like terms, the fit cannot distinguish an Odderon pole from a possible Pomeron-Odderon cut.
The result is a stability test of this effective amplitude decomposition, not a direct determination of a physical Odderon pole. The pole-versus-cut distinction remains unresolved within this setup.
The work is a preprint: arXiv version 1 [hep-ph], dated 26 August 2026. It was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico.
Paper data and sources
Original title: Odderon in the diffractive dip region
Authors: E. G. S. Luna, M. G. Ryskin, V. A. Khoze
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text