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

This paper investigates the form-factor dependence of Reggeized spin-2 Pomeron and spin-3 Odderon exchanges in high-energy elastic $pp$ and $p\bar p$ scattering, demonstrating that an exponential Odderon-proton form factor provides a superior fit to global data compared to other parametrizations and reveals a peripheral soft Odderon interaction with hadronic-scale transverse structure.

The Gist

Imagine two tiny, incredibly fast billiard balls (protons) zooming toward each other at nearly the speed of light. When they miss each other by a hair's breadth and bounce off, they don't just scatter randomly. They leave behind a specific pattern of "splatter" on the wall, which physicists call a differential cross-section.

For decades, scientists have tried to predict exactly what this splatter pattern looks like. They know that when protons bounce off other protons ($pp$), the pattern looks slightly different than when a proton bounces off an anti-proton ($p\bar{p}$). This difference is the "smoking gun" of a mysterious force carrier called the Odderon.

Think of the Pomeron as a friendly, invisible handshake that both protons and anti-protons feel the same way. It's the main reason they bounce. The Odderon, however, is like a grumpy ghost that only interacts differently with matter versus anti-matter. It's the reason the bounce patterns aren't identical.

The Problem: How "Soft" is the Ghost?

The paper asks a specific question: What does this "grumpy ghost" (the Odderon) actually look like?

In physics, particles aren't just hard dots; they have a "fuzziness" or a shape, described by something called a form factor. Imagine trying to describe the shape of a cloud. Is it a perfect sphere? A flat pancake? A jagged rock?

• Some scientists thought the Odderon was shaped like a dipole (a specific mathematical curve, like a bell curve).
• Others thought it might be a polynomial (a complex, wavy shape).
• Some guessed it was a Gaussian (a smooth, bell-shaped curve).

The authors of this paper decided to test seven different shapes (mathematical formulas) to see which one best describes how the Odderon interacts with the proton.

The Experiment: A Shape-Shifting Contest

The researchers took a massive amount of data from real-world experiments (from the LHC in Europe and the Tevatron in the US). They ran a simulation where they tried to fit these seven different "Odderon shapes" to the real data.

Think of it like trying to find the right key for a lock. You have seven different keys (the seven shapes). You try them all against the lock (the data).

• Keys 1 through 6: These were like keys that were okay. They fit, but they were a bit loose. They gave a "good enough" result, but not perfect.
• Key 7 (The Exponential): This key fit perfectly. It was the only one that turned the lock smoothly without any grinding.

The Big Discovery

The paper found that the Odderon isn't a jagged rock or a complex wavy shape. It behaves like a smooth, exponential curve.

Here is the cool part: In physics, a smooth exponential curve in momentum space translates to a Gaussian (bell-shaped) cloud in physical space.

• The Metaphor: If you could take a snapshot of the Odderon as it hits the proton, it wouldn't look like a sharp spike hitting the center. Instead, it looks like a soft, fuzzy cloud that gently brushes the outer edges (the periphery) of the proton.
• The authors calculated the size of this "fuzzy cloud" and found it to be about the size of a proton, but slightly larger, suggesting the Odderon interacts with the "skin" of the proton rather than its core.

Why Does This Matter?

1. It Solves a Puzzle: For years, models struggled to explain the tiny differences between proton-proton and proton-anti-proton collisions. By using this specific "smooth cloud" shape (the exponential form factor), the math finally matches the real-world data almost perfectly.
2. It Reveals the "Fuzziness": The study confirms that the Odderon is a "soft" interaction. It doesn't punch the proton in the nose; it gently grazes the side.
3. The Energy Limit: The authors noticed something interesting. At lower energies, this "smooth cloud" model works for a wide range of angles. But at the highest energies (like the 13 TeV at the LHC), the model starts to break down at wider angles.
   • The Analogy: Imagine trying to predict the path of a leaf in a gentle breeze (low energy). You can do it easily. But if the wind turns into a hurricane (high energy), the leaf starts doing crazy things the simple model can't predict. This suggests that at high energies, the protons start "absorbing" the collision or interacting in more complex ways that this simple model doesn't yet capture.

Summary

The paper is essentially a "shape contest" for a mysterious particle called the Odderon. After testing seven different mathematical shapes against real experimental data, the authors found that the Odderon is best described as a smooth, exponential cloud that gently grazes the outside of the proton. This simple shape explains the data better than any complex alternative, giving physicists a clearer picture of how these subatomic particles interact at the highest energies.

Technical Summary

Technical Summary: Odderon Form Factors in Reggeized Spin-2 Pomeron and Spin-3 Odderon Exchange

Problem Statement
High-energy elastic proton-proton ($pp$) and proton-antiproton ($p\bar{p}$) scattering serves as a critical probe for the non-perturbative dynamics of Quantum Chromodynamics (QCD). While Regge theory successfully describes the dominant $C=+1$ Pomeron exchange, persistent systematic differences between $pp$ and $p\bar{p}$ differential cross-sections, particularly in the dip–bump region, necessitate the inclusion of a leading $C=-1$ exchange: the Odderon. Previous phenomenological models utilizing spin-2 Pomeron and spin-3 Odderon exchanges have improved the description of differential cross-sections but have struggled to simultaneously reproduce central data values across all energy scales. A significant source of theoretical uncertainty in these models is the functional dependence of the Odderon–proton vertex, specifically the hadronic form factor. The precise nature of this form factor—whether it follows dipole, polynomial, Gaussian, or exponential behaviors—remains an open question, and its impact on the extraction of Regge parameters and the interpretation of the transverse spatial structure of the Odderon requires systematic investigation.

Methodology
The authors extend the tensor Pomeron–Odderon framework by systematically decoupling the spin structure from the Reggeized scalar kernels. This factorization allows for the isolation of form-factor dependence from trajectory dynamics. The study employs explicit covariant spin-2 and spin-3 projectors for the Pomeron and Odderon, respectively, while treating the Regge propagators as scalar functions.

The core of the analysis involves testing seven distinct parametrizations for the Odderon–proton form factor ($F_O(t)$) against a global dataset comprising:

• TOTEM data: $pp$ scattering at $\sqrt{s} = 2.76, 7, 8,$ and $13$ TeV.
• Tevatron (DØ) data: $p\bar{p}$ scattering at $\sqrt{s} = 1.80$ and $1.96$ TeV.

The seven tested form factors (FF1–FF7) include:

1. Standard dipole (FF1).
2. Quartic dipole (FF2).
3. Generalized polynomial series (FF3).
4. Gaussian (FF4).
5. Hybrid Gaussian–generalized dipole (FF5).
6. Hybrid Gaussian–polynomial (FF6).
7. One-parameter exponential $t$-slope: $F_O(t) = \exp[-B|t|/2]$ (FF7).

The scattering amplitudes are constructed using effective Lagrangians for spin-2 and spin-3 exchanges, Reggeized to satisfy unitarity and crossing symmetry via signature factors. The differential cross-sections are calculated by summing over spins and accounting for the interference between the Pomeron and Odderon amplitudes. A global fit is performed using Monte Carlo error propagation (parametric bootstrap) to determine best-fit parameters and statistical uncertainties. The analysis restricts the $|t|$ range to the "single-Regge-pole validity window," excluding the Coulomb-nuclear interference region and the high-$|t|$ region where absorptive corrections become dominant.

Key Contributions

1. Factorized Framework: The paper introduces a formalism that separates the covariant tensor projectors from the Reggeized scalar propagators, enabling a clean comparison of vertex form factors without confounding trajectory parameter shifts.
2. Systematic Form-Factor Scan: It provides the first systematic comparison of seven distinct Odderon–proton form-factor parametrizations within a coherent tensor-exchange framework.
3. Identification of a Preferred Form: The study identifies a clear statistical hierarchy among the models, demonstrating that a simple one-parameter exponential form factor provides a superior description of the data compared to more complex dipole or polynomial structures.
4. Geometric Interpretation: The authors map the momentum-transfer dependence of the optimal form factor to impact-parameter space, interpreting the exponential slope as a Gaussian transverse profile.

Results

• Fit Quality: The exponential form factor (FF7) yields a reduced chi-squared of $\chi^2_{\text{red}} \simeq 0.98$ for 138 degrees of freedom, a significant improvement over the six other parametrizations (FF1–FF6), which cluster around $\chi^2_{\text{red}} \simeq 1.44–1.48$.
• Parameter Stability: The fitted couplings ($g_{PNN} \approx 5.61$, $g_{ONN} \approx 13.83$) and Regge slopes remain comparatively stable across the different form-factor choices. This indicates that the improvement in fit quality is driven primarily by the Odderon–proton vertex shape rather than by compensating shifts in trajectory parameters.
• Transverse Structure: The exponential form factor $F_O(t) = \exp[-B|t|/2]$ corresponds to a Gaussian profile in impact-parameter space, $\tilde{F}(b) \propto \exp[-b^2/(2B)]$. The extracted effective radii $\sqrt{\langle b^2 \rangle} = \sqrt{2B}\hbar c$ are of hadronic size and generally exceed the proton charge radius ($\sim 0.84$ fm), suggesting a peripheral soft Odderon interaction.
• Energy Dependence and Validity Window: The analysis reveals a contraction of the $|t|$ range over which the single-Regge-pole description remains accurate as energy increases. At $\sqrt{s} = 2.76$ TeV, the model fits data up to $|t| \approx 0.6$ GeV$^2$, whereas at $\sqrt{s} = 13$ TeV, deviations appear around $|t| \approx 0.20$ GeV$^2$. This contraction is interpreted as the onset of absorptive corrections and unitarity effects (e.g., multi-Pomeron cuts) at LHC energies.
• Mass Parameter: The fitted Odderon mass parameter $M_O$ varies significantly between $pp$ and $p\bar{p}$ subsets (e.g., $\sim 2.8$ GeV for $pp$ vs. $\sim 0.1$ GeV for $p\bar{p}$). The authors caution that this should not be interpreted as a physical glueball mass but rather as an effective kernel scale within the restricted kinematic windows.

Significance and Claims
The paper claims to provide a compact phenomenological framework that connects the $pp/p\bar{p}$ dip–bump difference with the transverse structure of $C$-odd color-singlet exchange. Its primary significance lies in demonstrating that the soft Odderon interaction is best described by a smooth, exponential vertex corresponding to a Gaussian transverse profile, rather than the power-law behaviors often assumed in dipole models.

The authors modestly frame their results as an "effective" description. They explicitly state that the extracted geometric scales are model-dependent effective transverse scales rather than uniquely determined proton observables, due to parameter correlations and the lack of a full unitarization treatment. The work serves as a starting point for future eikonalized Regge analyses and offers a benchmark for comparisons with non-perturbative QCD descriptions, including holographic-QCD approaches, but does not claim to definitively determine the fundamental mass of the Odderon or the exact nature of the underlying QCD dynamics without further constraints from total cross-sections and forward amplitude analyses.