Constraints on the odderon amplitude in the CGC framework

This paper analyzes elastic $pp$ and $p\bar{p}$ scattering within the Color Glass Condensate framework using experimental data to derive phenomenological upper bounds on the odderon amplitude, finding that while current parametrizations can describe the data, large uncertainties limit the precision of these constraints and highlight the need for improved measurements.

The Gist

Here is an explanation of the paper, translated into everyday language using analogies.

The Big Picture: Hunting for a Ghost Particle

Imagine the universe is a giant dance floor where particles are constantly bumping into each other. Physicists have known for decades about a "dance partner" called the Pomeron. It's like a popular, reliable DJ who plays the same beat over and over, causing protons (the main dancers) to bounce off each other in predictable ways.

But there is a rumor of a second, much more elusive partner called the Odderon.

• The Pomeron is like a handshake: it treats matter (protons) and antimatter (antiprotons) exactly the same.
• The Odderon is like a high-five that only works one way: it treats matter and antimatter differently.

For 50 years, scientists have been trying to spot this "Odderon" ghost. Recently, two big experiments (TOTEM at the LHC and D0 at the Tevatron) claimed they finally saw a mismatch in how protons and antiprotons bounce off each other. They said, "Aha! The Odderon is here!"

This paper asks a critical question: Is that mismatch really the Odderon, or is it just a trick of the light (or a measurement error)?

The Setup: The "Dilute-Dense" Sandwich

To figure this out, the authors used a theoretical framework called the Color Glass Condensate (CGC). Think of this as a way to model the proton not as a solid ball, but as a cloud of fog.

• The Projectile (The Bullet): One proton is like a thin, wispy cloud of fog (a "dilute" system) flying at high speed.
• The Target (The Wall): The other proton is like a dense, thick wall of fog (a "dense" system).

When the thin cloud hits the thick wall, it scatters. The authors calculated exactly how this scattering should look if the "Odderon" ghost is present versus if it isn't.

The Detective Work: Building a Special Filter

The problem is that the "Odderon" signal is tiny. It's like trying to hear a whisper (the Odderon) while a loud rock band (the Pomeron) is playing right next to you. The rock band drowns out the whisper.

To solve this, the authors invented a mathematical filter (an observable called $\Sigma$).

• Imagine you have two microphones: one recording the "Proton vs. Proton" collision, and one recording the "Proton vs. Antiproton" collision.
• The loud rock band (Pomeron) sounds the same on both microphones.
• The whisper (Odderon) sounds different on them.

By subtracting the two recordings, the loud rock band cancels out, leaving only the whisper. The authors created a formula that does exactly this subtraction, isolating the "Odderon" signal from the background noise.

The Experiment: Testing Two "Ghost" Theories

The authors took this filter and applied it to real data from the TOTEM and D0 experiments. They tested two different theories about what the Odderon looks like:

1. Theory A (The Jeon-Venugopalan Model): This theory suggests the Odderon has a specific shape, like a specific type of fog pattern that depends on how far the particles are from the center of the collision.
2. Theory B (The Hatta-Iancu Model): This theory suggests the Odderon is more like a spinning top, depending on the angle and spin of the collision.

They ran the numbers, adjusting the "volume" (normalization) of the Odderon to see if they could make the theory match the data.

The Results: The Ghost is Still Elusive

Here is what they found:

• The Data is Noisy: The experimental data has huge "error bars." Imagine trying to measure the height of a person while they are standing on a trampoline. The measurements jump around a lot.
• The "Ghost" is Weak: Even with their special filter, the data didn't scream "Odderon!" It just whispered. The mismatch between protons and antiprotons could be explained by the "loud rock band" (Pomeron) playing slightly differently than expected, without needing a ghost at all.
• The Constraints: The authors found that if the Odderon were as strong as some people claimed to explain the TOTEM-D0 data, it would break the laws of physics (it would predict other collisions happening way too often, which we don't see).
• The Verdict: The current data is too fuzzy to prove the Odderon exists. The "mismatch" seen in the experiments is likely just a combination of measurement errors and the fact that different experiments (TOTEM and D0) might have slightly different calibration settings.

The Takeaway

Think of this paper as a reality check.

While the idea of the Odderon is exciting and theoretically sound, the authors are saying: "Don't celebrate yet. The evidence we have right now is too blurry to be sure. The 'ghost' we think we see might just be a shadow cast by our own measuring tools."

They conclude that we need better microphones (more precise experiments) and new ways to listen (different types of collisions, perhaps involving polarized beams or future colliders like the Electron-Ion Collider) before we can definitively say the Odderon is real.

In short: The Odderon is a fascinating theoretical possibility, but the current evidence is too weak to confirm its existence, and the "signals" we see might just be noise.

Technical Summary

Here is a detailed technical summary of the paper "Constraints on the odderon amplitude in the CGC framework" by Roa et al.

1. Problem Statement

The Odderon is a charge-parity (C-odd) exchange counterpart to the well-established Pomeron (C-even) in Quantum Chromodynamics (QCD). While predicted theoretically in the 1970s, its experimental confirmation has remained elusive due to the difficulty of isolating its signal from the dominant C-even Pomeron background.

• Current Status: Recent comparisons of proton-proton ($pp$) and proton-antiproton ($p\bar{p}$) elastic scattering data (specifically from TOTEM and D0 collaborations) suggest a discrepancy that could be attributed to the Odderon. However, the statistical significance of this signal is debated, and theoretical uncertainties regarding the normalization of the Odderon amplitude are large.
• The Challenge: Existing phenomenological models often rely on arbitrary normalization parameters. Furthermore, extracting the Odderon signal is complicated because it is commingled with large C-even contributions and electromagnetic (photon) exchanges in the cross-section.
• Goal: The authors aim to derive phenomenological constraints (upper bounds) on the Odderon amplitude within the Color Glass Condensate (CGC) framework and propose a robust observable to minimize uncertainties associated with the C-even sector.

2. Methodology

Theoretical Framework

The analysis is conducted within the CGC effective field theory, treating high-energy $pp$ and $p\bar{p}$ collisions as a dilute-dense system.

• Proton Structure: The proton is modeled as a valence three-quark Fock state. The authors utilize two distinct non-perturbative light-front wave functions to assess model dependence:
  1. A Gaussian parametrization (Ref. [28, 29]), adjusted to match the experimental proton RMS radius.
  2. An AdS/QCD-based parametrization (Ref. [31]), derived from light-front holography.
• Scattering Amplitude: The interaction is described by Wilson line correlators. The dipole scattering amplitude is decomposed into C-even (Pomeron, $N$) and C-odd (Odderon, $O$) parts:
  $$\frac{1}{N_c} \langle \text{tr}(U U^\dagger) \rangle = 1 - N(Y, r, b) + iO(Y, r, b)$$
  The evolution of these amplitudes with rapidity $Y$ is governed by the BK-JIMWLK equations. The authors solve the coupled evolution equations numerically, noting that in the limit $O \ll N$, the Odderon evolution becomes linear and decoupled from the Pomeron.

Parametrizations of the Odderon

Two popular phenomenological parametrizations for the initial Odderon amplitude $O(Y_0, r, b)$ are tested:

1. Jeon-Venugopalan (JV) Model (Ref. [16]): Incorporates transverse geometry via a Gaussian profile and a local saturation scale. It depends on the dipole size $r$ and impact parameter $b$.
2. Spin-Dependent Model (Ref. [12, 23]): Originally dependent on the target spin, this is adapted for unpolarized observables by introducing an impact-parameter dependent profile $S(b)$ and a factor $(\hat{r} \cdot \hat{b})$ to satisfy antisymmetry.

The Proposed Observable

To isolate the Odderon contribution, the authors construct a specific observable $\Sigma$ based on the difference and sum of differential cross-sections:
$$\Sigma = \frac{d\sigma(pp)/dt - d\sigma(p\bar{p})/dt}{\sqrt{d\sigma(pp)/dt + d\sigma(p\bar{p})/dt}}$$

• Rationale: In the approximation where the Odderon amplitude is small ($|O| \ll |N|$), the sum of cross-sections is dominated by the C-even Pomeron ($|N|^2$), while the difference is proportional to the interference term ($\text{Re}(N^* O)$). This observable effectively cancels the dominant C-even uncertainty and isolates the C-odd contribution (plus a known photon term).

3. Key Contributions

1. Novel Observable: The definition of $\Sigma$ allows for a direct comparison between theory and experiment that is largely immune to uncertainties in the C-even Pomeron amplitude normalization.
2. Systematic Model Testing: The study rigorously tests the sensitivity of the results to:
   • Different proton wave functions (Gaussian vs. AdS/QCD).
   • Different impact-parameter profiles ($S(b)$).
   • Different Odderon parametrizations.
3. Global Fit Analysis: The authors perform a global $\chi^2$ minimization to determine the normalization factor $\lambda$ (scaling the Odderon amplitude) that best fits the experimental data from ISR (Intersecting Storage Rings) and TOTEM-D0.

4. Results

• Data Consistency: The ISR and TOTEM-D0 datasets show only marginal consistency in the kinematic region $|t| < 0.7 \text{ GeV}^2$. The authors find it difficult to find a single parametrization that fits the smooth $t$-dependence of ISR data and the "node" structure observed in D0 data simultaneously.
• Fit Quality:
  • When fitting the data with the proposed observable, the inclusion of the Odderon term (with a free normalization $\lambda$) does not significantly improve the $\chi^2$ compared to a photon-only contribution.
  • The best-fit values for the normalization $\lambda$ are large (order of 100) when fitting only TOTEM-D0 data, but this leads to unphysical cross-sections for exclusive processes (e.g., $\pi^0, \eta_c$ production) and violates theoretical unitarity constraints (Eq. 33 in the paper).
• Constraints:
  • ISR Data Dominance: The most stringent upper limits on the Odderon amplitude come from the ISR data at large momentum transfer ($|t| > 1 \text{ GeV}^2$).
  • TOTEM-D0 Tension: Interpreting the TOTEM-D0 mismatch as a pure Odderon signal requires an amplitude that is theoretically inconsistent with other known constraints (such as HERA limits on exclusive production).
  • Uncertainty: Due to large experimental errors, the Odderon amplitude remains "loosely constrained." The data allows for a range of normalizations, but the central values suggest the signal is small or non-existent within the tested frameworks.
• Wave Function Independence: The choice of proton wave function (Gaussian vs. AdS/QCD) primarily affects the global normalization of the predicted signal but has a negligible effect on the shape of the $t$-dependence.

5. Significance and Conclusions

• Limited Sensitivity: The study concludes that current experimental data (ISR and TOTEM-D0) provide limited sensitivity to the Odderon. The large systematic uncertainties and the dominance of the C-even background prevent a definitive extraction of the Odderon amplitude.
• Theoretical Tension: The authors argue that the "compelling evidence" for the Odderon from the TOTEM-D0 comparison is likely an overestimation. Attributing the observed mismatch solely to the Odderon requires amplitudes that violate theoretical bounds and contradict other experimental limits (e.g., HERA).
• Future Directions: The paper emphasizes the need for:
  • Improved Precision: Higher precision data to reduce the large error bars currently masking the signal.
  • Complementary Observables: The authors suggest that spin-sensitive polarized observables and odderon-mediated quarkonia photoproduction (e.g., at the future Electron-Ion Collider, EIC) offer more promising avenues for isolating the C-odd exchange without the ambiguities present in inclusive elastic scattering.

In summary, while the paper provides a robust formalism for analyzing C-odd exchanges in the CGC framework, it casts doubt on the strength of the current Odderon signal claims, suggesting that existing data is insufficient to tightly constrain the Odderon amplitude and that the observed discrepancies may stem from systematic normalization issues rather than new physics.