The eikonal spin-dependent Odderon and gluon Sivers function of a proton, and its small-$x$ evolution

This paper utilizes a three-quark light-front model to calculate the gluon Sivers function of a proton at moderately small $x$ and numerically determines its small-$x$ evolution via the BFKL anomalous dimension, revealing a power-law tail behavior of $k_\perp^{-3.3}$ at high transverse momentum.

The Gist

Here is an explanation of the paper "The eikonal spin-dependent Odderon and gluon Sivers function of a proton, and its small-x evolution," translated into everyday language with creative analogies.

The Big Picture: Spinning a Top in a Storm

Imagine a proton not as a solid marble, but as a tiny, chaotic storm cloud made of three main dancers (quarks) and a swirling sea of invisible energy (gluons).

Physicists have long known that if you spin this proton (like a top), the particles inside don't just spin with it; they also drift sideways. This sideways drift is called the Sivers Effect. It's like if you spun a figure skater, and their arms suddenly started flailing to the left or right depending on how fast they were spinning.

This paper is about measuring exactly how that sideways drift happens, specifically for the "gluons" (the energy glue holding the proton together), and figuring out how this behavior changes when we look at the proton under extreme conditions (very high energy).

The Main Characters

1. The Proton: Our main character, a spinning top made of quarks and gluons.
2. The Sivers Function: A map that tells us: "If the proton spins this way, the gluons are most likely to be found drifting that way."
3. The Odderon: A ghostly, invisible force that appears when particles smash together at high speeds. Think of it as a "shadow" cast by the proton. Usually, shadows are boring (symmetrical), but the Spin-Dependent Odderon is a special, twisted shadow that only appears when the proton is spinning. It's the "ghost in the machine" that causes the sideways drift.
4. The "Small-x" Evolution: Imagine zooming in on the proton. As you zoom in (or as the proton moves faster), you see more and more tiny gluons popping in and out of existence. "Evolution" is the math that describes how the proton's internal storm changes as you zoom in deeper.

The Story of the Paper

1. Building a Model of the Proton (The "Three-Dancer" Model)

The authors start by building a model of the proton. They imagine it as three quarks dancing on a stage. To make the model realistic, they use a "Light-Front Wave Function."

• Analogy: Think of a photograph of a spinning dancer. A normal photo is blurry. This model is like a high-speed camera that freezes the dancer's pose, showing exactly where their feet are and how their arms are positioned.
• The Twist: The authors realized that for the proton to spin and create this sideways drift, the quarks can't just spin in place. They have to use their Orbital Angular Momentum.
• Metaphor: Imagine a planet orbiting a star. If the planet spins on its axis and orbits the star, it creates a complex motion. The authors found that the quarks are "orbiting" inside the proton. When the proton spins, this orbital motion is what pushes the gluons sideways.

2. The "Ghost" Calculation (The Odderon)

The team calculated how this spinning proton interacts with a passing particle. They found that the interaction creates a "C-odd" signal (the Odderon).

• The Discovery: They found that the "ghost" (the Odderon) is strongest when the gluons are moving at a specific speed (around 0.5 GeV).
• The Shape: The map of this sideways drift (the Sivers function) looks like a hill. It starts low, goes up to a peak (where the drift is strongest), and then falls off.
• The Surprise: In some older theories, this map was expected to be a smooth hill starting from zero. But this paper shows that at very low speeds, the map actually spikes or "diverges" (gets very large). It's like a hill that suddenly has a sharp, jagged peak at the very bottom.

3. The Evolution (The "Zoom" Effect)

Next, they asked: "What happens if we zoom in even further?" (This is the BFKL evolution).

• The Process: As the proton moves faster and faster, more gluons are created. The team used a complex mathematical equation (the BFKL equation) to simulate this.
• The Result: As they zoomed in, the "hill" of the Sivers function changed shape. The peak stayed, but the "tail" of the hill (the part where the drift is weak) became steeper.
• The Power Law: They found a specific rule for how the drift fades away at high speeds. It fades away like $1/k^{3.3}$.
  • Analogy: Imagine throwing a stone into a pond. The ripples get smaller as they move away. The authors calculated exactly how fast those ripples die out. They found that for this spinning proton, the ripples die out faster than usual because of the complex three-gluon dance happening inside.

Why Does This Matter?

1. It's a New Map: Before this, we didn't have a good map of how gluons drift sideways in a spinning proton. This paper provides a detailed map based on a realistic model of the proton's interior.
2. It Explains "Cancellations": The authors found that if you add up all the sideways drifts, they almost cancel each other out.
   • Analogy: Imagine a room full of people pushing a heavy box. Some push left, some push right. If you just look at the total force, it might look like nothing is happening. But if you look at the individual people, you see a lot of effort. This explains why some experiments haven't seen the effect yet—the pushes are canceling out.
3. Future Experiments: This research is crucial for the Electron-Ion Collider (EIC), a massive new machine being built to smash electrons into protons. The authors are saying, "Here is what you should look for. If you see a peak at 0.5 GeV and a specific fading pattern, you've found the Spin-Dependent Odderon!"

The Bottom Line

This paper is like a detective story. The authors built a model of a spinning proton, calculated the "ghostly" forces inside it, and predicted exactly how those forces should look when we zoom in. They found that the proton's internal dance is more complex than we thought, involving orbital motion and specific "ghost" forces that will be the next big thing for physicists to discover at the new Electron-Ion Collider.

In one sentence: They figured out how the "glue" inside a spinning proton drifts sideways, discovered it has a specific peak and a unique fading pattern, and provided a roadmap for future scientists to find it.

Technical Summary

Here is a detailed technical summary of the paper "The eikonal spin-dependent Odderon and gluon Sivers function of a proton, and its small-x evolution" by Benić et al.

1. Problem Statement

The paper addresses the theoretical understanding of the gluon Sivers function ($xf_{1T}^{\perp g}(x, k_\perp)$) of the proton, a Transverse Momentum Dependent (TMD) parton distribution function (PDF) that describes the correlation between the proton's transverse spin and the transverse momentum of gluons.

• Current Gap: While the gluon Sivers function is a primary target for the future Electron-Ion Collider (EIC), it is poorly constrained experimentally and theoretically. Existing models (e.g., D'Alesio et al.) often lack a rigorous connection to the underlying QCD dynamics at small Bjorken-$x$.
• Theoretical Link: In the eikonal approximation, the gluon Sivers function corresponds to the C-odd component of the dipole scattering amplitude, known as the spin-dependent Odderon.
• Specific Challenge: Calculating the Odderon requires a non-zero matrix element with a helicity flip ($\Lambda' = -\Lambda$) in the forward limit. Standard perturbative calculations often struggle with this, and previous models (like the extended McLerran-Venugopalan model) relied on high-dimensional color/spin representations. The authors aim to derive this from a more fundamental, effective three-quark light-front model where sources are in the fundamental representations of color and spin.

2. Methodology

The authors employ a multi-stage approach combining non-perturbative modeling with perturbative QCD evolution:

• Light-Front (LF) Three-Quark Model:

  • They utilize an effective three-quark light-cone wave function (LCwf) for the proton, valid for moderate $x \sim 0.1$ and transverse momenta $k_\perp \lesssim 1$ GeV.
  • The wave function incorporates Melosh rotations, which are crucial for generating the necessary orbital angular momentum (OAM) to flip the proton's helicity without flipping individual quark helicities (since the eikonal operator conserves quark helicity).
  • The model is validated against experimental data for the proton's Dirac and Pauli electromagnetic form factors.

• Eikonal Odderon Calculation:

  • The spin-dependent Odderon is calculated as the expectation value of the C-odd cubic Casimir operator (three-gluon exchange) between proton states of opposite helicity.
  • The calculation explicitly demonstrates that a helicity flip in the forward limit requires the three exchanged gluons to attach to three different quarks in the proton. Contributions where gluons attach to one or two quarks vanish in the forward limit for helicity-flip processes.

• Small-$x$ Evolution (BFKL):

  • The non-perturbative LCwf result serves as the initial condition at $x_0 \sim 0.1$.
  • The authors solve the linear BFKL evolution equation (in the Mueller dipole framework) to evolve the Odderon to smaller $x$ (higher rapidity $Y = \ln(x_0/x)$).
  • They analyze the emergence of an anomalous dimension characterizing the power-law tail of the Sivers function at high $k_\perp$.

3. Key Contributions

• Fundamental Derivation: The paper provides the first derivation of the eikonal spin-dependent Odderon and the associated gluon Sivers function using a three-quark light-front model with sources in the fundamental representations of $SU(3)_c$ and $SU(2)_s$.
• Mechanism of Helicity Flip: It clarifies that the proton helicity flip in the forward limit is mediated by quark orbital angular momentum (OAM) transfer, specifically requiring a three-gluon coupling to three distinct quarks.
• Low-$k_\perp$ Behavior: The authors identify a logarithmic divergence in the Sivers function at very low transverse momenta ($k_\perp \to 0$), contrasting with previous models that predicted a power-law decay or a finite constant. This has implications for observables integrating over $k_\perp$.
• BFKL Anomalous Dimension: They numerically determine the anomalous dimension $\gamma(Y)$ governing the power-law tail of the Sivers function at small $x$, finding $\gamma \approx 0.8$ at $Y=4$.

4. Key Results

A. The Gluon Sivers Function at $x_0 \sim 0.1$

• Magnitude and Shape: The Sivers function peaks at $k_\perp \approx 0.5$ GeV with a magnitude of $(4-5) \times 10^{-3}$ GeV$^{-2}$.
• Sign Definiteness: The function is not sign-definite; it changes sign, leading to substantial cancellations in its first moment (the twist-3 collinear tri-gluon PDF).
• Phenomenological Consequence: Due to these cancellations, the ratio of the Odderon exchange cross-section to the Primakoff (photon exchange) cross-section in exclusive forward $\chi_{c1}$ photoproduction is extremely small ($r \approx 10^{-6}$). This suggests that photon exchange dominates this process at $x \sim 0.1$, contrary to some optimistic estimates.
• Parameterization: The authors provide a fit formula for the Sivers function:
  $$k_\perp x f_{1T}^{\perp g} \propto \frac{k_\perp}{Q_b} \ln\left(\frac{k_\perp}{Q_b}\right) e^{-k_\perp^2 / (4Q_c^2)}$$
  with parameters $Q_b \approx 0.33$ GeV and $Q_c \approx 0.22$ GeV.

B. Small-$x$ Evolution (BFKL)

• Radial Behavior: As rapidity $Y$ increases, the Odderon amplitude $O_S(Y, r_\perp)$ grows at small dipole sizes ($r_\perp \lesssim 0.45$ fm) but is suppressed at larger sizes.
• Anomalous Dimension: The evolution generates an anomalous dimension $\gamma(Y)$.
  • At $Y=0$ (initial condition), $\gamma = 1$, leading to a $k_\perp^{-4}$ tail.
  • At $Y=4$ (smaller $x$), $\gamma$ decreases to $\approx 0.8$.
  • Consequently, the high-$k_\perp$ tail of the Sivers function evolves from $\sim k_\perp^{-4}$ to $\sim k_\perp^{-3.2}$ (specifically $k_\perp^{-4\gamma}$).
• Comparison: This power-law tail is steeper than the unpolarized gluon TMD ($\sim k_\perp^{-2}$) due to the three-gluon nature of the Odderon, similar to diffractive TMDs.

5. Significance

• EIC Physics: The results provide a rigorous, QCD-based initial condition for the gluon Sivers function, essential for interpreting future EIC data. The prediction of a peak at $k_\perp \sim 0.5$ GeV (rather than at zero) offers a distinct signature for experimental verification.
• Theoretical Consistency: The work bridges the gap between non-perturbative constituent quark models and perturbative small-$x$ evolution, demonstrating how OAM in the proton wave function drives the spin-dependent Odderon.
• Phenomenology: The finding that the tri-gluon PDF is heavily suppressed by cancellations implies that forward exclusive $\chi_{c1}$ production is likely dominated by photon exchange, refining predictions for background processes in heavy quarkonium production.
• Open Data: The authors have made their numerical data and code publicly available, promoting reproducibility in the field of high-energy QCD.

In summary, this paper establishes a robust theoretical framework for the gluon Sivers function, revealing its complex structure (logarithmic divergence at low $k_\perp$, sign-changing behavior, and specific power-law tails) and quantifying its evolution from the non-perturbative regime to the small-$x$ limit.