⚛️ phenomenology

Odderon in the diffractive dip region

This paper analyzes collider data on elastic proton-proton and proton-antiproton scattering in the diffractive dip region using a Regge-inspired parametrization and finds that the fits do not require a stable, nonzero $C$ -odd Odderon contribution, as the data are well-described by $C$ -even terms alone or remain inconclusive when additional Odderon terms are introduced.

Original authors: E. G. S. Luna, M. G. Ryskin, V. A. Khoze

Published 2026-08-27

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CC BY 4.0

Original authors: E. G. S. Luna, M. G. Ryskin, V. A. Khoze

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles like protons do not bounce off one another like billiard balls; instead, they interact through a complex exchange of invisible forces that can be thought of as a fog of energy. When two protons collide and scatter without breaking apart, a process known as elastic scattering, the pattern of their deflection reveals the structure of this interaction. Physicists have long understood that most of this interaction is driven by a dominant, symmetric force called the Pomeron, which acts like a smooth, predictable tide. However, theory also predicts a much rarer, asymmetrical counterpart known as the Odderon. This Odderon is a ghostly, negative-symmetry component that should exist but is expected to be so faint that it is easily drowned out by the overwhelming presence of the Pomeron. Finding it is a major goal in particle physics because its discovery would confirm a specific prediction of the theory governing the strong nuclear force, the glue that holds atomic nuclei together.

For decades, scientists have searched for signs of this Odderon by comparing how protons scatter against other protons versus how they scatter against antiprotons. The hope is that the Odderon would create a subtle difference in the scattering patterns, particularly in a specific region where the probability of scattering drops sharply, creating a dip in the data. A recent study by researchers in Brazil, Russia, and the United Kingdom took a fresh, direct look at this problem. They gathered data from high-energy collisions at major particle accelerators, including the Large Hadron Collider and the Tevatron, covering a wide range of energies from 546 billion electron volts up to 13 trillion electron volts. Their goal was to see if the data in the "dip region" required the presence of an Odderon to be explained, or if the known forces were sufficient on their own.

The team constructed a detailed mathematical model to describe the scattering data, treating the interaction as a sum of different contributing forces. They started by modeling the dominant, symmetric part of the interaction using three distinct, effective components that behave like the Pomeron. This flexible approach allowed them to capture the complex shape of the scattering data without forcing it into a rigid, pre-defined box. Once this strong, symmetric background was established, they tested whether adding an Odderon component improved the fit. They tried several variations: first assuming the Odderon had a fixed, simple shape, then allowing its properties to vary more freely, and finally testing if adding a second, different Odderon-like term was necessary.

The results were surprisingly clear. When the researchers allowed the symmetric part of the interaction enough freedom to describe the data accurately, the data did not demand the presence of a stable, non-zero Odderon. In the simplest models, the contribution from the Odderon was consistent with zero. When they gave the Odderon more freedom to adjust its shape, the resulting contribution remained very small and unstable, meaning the data could not pin down a specific, reliable value for it. The researchers found that the improvements they saw in their models came almost entirely from refining the description of the symmetric, Pomeron-like forces, not from adding the asymmetrical Odderon. Even when they added a fourth Pomeron-like term to make the symmetric part even more flexible, the fit improved, but adding a second Odderon-like term did not provide any comparable benefit. The data simply could not distinguish between two different odd components, nor did it show a clear need for one.

The study also highlighted some tensions in the existing data. While the model described the scattering data at 8 trillion electron volts very well, it struggled to fit the data from 7 trillion electron volts and the lower-energy antiproton data simultaneously. This suggests that the differences between the datasets might be due to experimental uncertainties or limitations in the simple model used, rather than a clear signal of a new particle. The researchers concluded that while the Odderon is not proven to be absent, the current data in the dip region does not provide a stable, independent confirmation of its existence when the dominant forces are allowed to vary freely. This finding offers a complementary perspective to other recent studies that used different mathematical assumptions to suggest the Odderon exists. Here, the direct approach shows that the complex structure of the scattering dip can be explained by the known symmetric forces alone, leaving the Odderon as a faint, elusive possibility that remains difficult to isolate from the background noise of the strong interaction.

Technical Summary: Odderon in the Diffraction Dip Region

Problem Statement
The paper investigates the existence and magnitude of the Odderon, the $C$ -odd (crossing-odd) component of the high-energy elastic scattering amplitude. While the Odderon is theoretically predicted in perturbative QCD as a three-gluon exchange, its experimental detection is challenging because it is expected to be significantly smaller than the dominant $C$ -even Pomeron exchange. The authors focus on the "diffractive dip" region in elastic proton-proton ($pp$) and proton-antiproton ( $\bar{p}p$ ) scattering. In this kinematic region, the imaginary part of the dominant even amplitude is suppressed, theoretically allowing a relatively small real odd amplitude to produce visible distortions in the differential cross-section ( $d\sigma/dt$ ). The central question is whether a stable, non-zero $C$ -odd contribution is required to describe collider data when the $C$ -even amplitude is given sufficient phenomenological freedom.

Methodology
The authors employ a direct phenomenological approach using a Regge-inspired parametrization, distinct from previous analyses that relied on scaling properties and analyticity constraints (e.g., Ref. [15]).

Amplitude Construction: The elastic amplitude $F(s,t)$ is decomposed into crossing-even ( $F_+$ ) and crossing-odd ( $F_-$ ) components. The total amplitude is modeled as a sum of effective terms with power-like energy dependence:
$F(s, t) = \sum_{i=1}^{N_P} F_{Pi}(s, t) + \tau \sum_{j=1}^{N_O} F_{Oj}(s, t)$
where $\tau = -1$ for $pp$ and $+1$ for $\bar{p}p$ .
Signature Factors: Crucially, the relative phases between the real and imaginary parts of the terms are not free parameters. They are strictly fixed by the signature factors ( $\eta_P$ for even, $\eta_O$ for odd), linking the energy dependence to the crossing phase.
Parametrization:
- The $C$ -even sector is initially described by three effective Pomeron-like terms (linear trajectories and exponential residues).
- The $C$ -odd sector is tested using one or two effective Odderon-like terms.
- The parametrization is treated as an effective description valid only within the fitted kinematic range ( $\sqrt{s} > 500$ GeV, $0.2 < |t| < 1$ GeV $^2$ ), not as an asymptotic amplitude or a representation of bare Regge poles.
Data Sets: The analysis fits 452 data points from collider experiments (UA4, CDF, E710, D0, TOTEM) covering energies from 546 GeV to 13 TeV.
Fit Strategy: Two data ensembles are used:
1. Ensemble A: Published experimental errors.
2. Ensemble B: A reweighted ensemble where 13 TeV systematic errors are inflated by a factor $f=3$ to test if conclusions are driven by the high statistical weight of the 13 TeV data.
Model Variants: The authors systematically test models (A1–A5) by:
- Fixing the Odderon trajectory ( $\alpha_O=1$ ) and residue slope (A1).
- Allowing the residue slope to vary (A2).
- Releasing the Odderon intercept and trajectory slope (A3).
- Adding a fourth Pomeron-like term to the even sector (A4).
- Adding a second Odderon-like term (A5).

Key Results
The fits yield the following findings regarding the $C$ -odd sector:

No Stable Non-Zero Odderon: In the baseline models (A1), the Odderon-like coupling is compatible with zero.
Weak Constraints with Freedom: When the Odderon sector is allowed more freedom (A2 and A3), the fits do not yield a stable, non-zero determination.
- In A2, a non-zero coupling is found, but the residue slope ( $B_O$ ) is large ( $\sim 8\text{--}9$ GeV $^{-2}$ ), causing the odd contribution to be exponentially suppressed in the dip region ( $|t| \approx 0.5$ GeV $^2$ ).
- In A3, the intercept and slope parameters are poorly constrained, with large uncertainties. The resulting odd amplitude in the dip region remains small (real part $\lesssim 3\text{--}4\%$ of the even amplitude).
Role of the Even Sector: The improvement in fit quality ( $\chi^2$ ) obtained by adding a fourth Pomeron-like term (Model A4) is significantly larger than that obtained by adding a second Odderon term (Model A5). This indicates that the structure of the dip region is primarily described by the flexibility of the effective $C$ -even amplitude.
Data Tensions: The analysis reveals tensions between data sets, particularly a large $\chi^2$ contribution from the 7 TeV data compared to the well-described 8 TeV data. The 13 TeV reweighting reduces the total $\chi^2$ but does not resolve discrepancies at lower energies (1.96 TeV, 2.76 TeV, 7 TeV).
Local Limitations: For $|t| \gtrsim 0.55$ GeV $^2$ , the models tend to undershoot both $pp$ and $\bar{p}p$ data. The authors note that increasing the $C$ -odd contribution would raise the $\bar{p}p$ cross-section but lower the $pp$ one, suggesting the missing contribution in this region must be primarily $C$ -even.

Significance and Claims
The paper concludes that within the framework of this effective Regge-inspired parametrization, a stable, non-zero $C$ -odd contribution is not required to describe the available collider data in the diffractive dip region.

Complementarity: The authors position this result as a complementary test to recent scaling-and-analyticity analyses (Ref. [15]). While the scaling analysis extracts a small Odderon by imposing analyticity on a scaling variable, this direct fit tests whether an odd component can be resolved independently when the even amplitude is given maximal phenomenological freedom. The result suggests that the even amplitude can absorb much of the dip-region structure, leaving little room for a distinct, stable odd term.
Limitations: The authors explicitly state that these conclusions do not prove the absence of the Odderon. Rather, they indicate that the current data and the specific effective parametrization (which cannot distinguish between an Odderon pole and a Pomeron-Odderon cut) are insufficient to resolve a stable odd contribution. The fit cannot separate a pole term from possible cut contributions, and the "Odderon-like" terms used are merely effective parametrizations.
Future Directions: The paper suggests that a more complete phenomenological description might require explicitly including cut contributions (e.g., $F_{cut}^{PO}$ ) alongside pole terms, which was not included in the current fits.

In summary, the paper provides a rigorous test of the Odderon hypothesis using flexible effective amplitudes, finding that the data does not necessitate a stable odd term and that the observed dip structures are well-accounted for by the $C$ -even sector alone.

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