Back-to-back dijet production in DIS with finite-energy corrections and twist-3 gluon TMDs

This paper calculates the next-to-eikonal cross section for back-to-back dijet production in deep inelastic scattering at small $x$, demonstrating how these corrections relate to the $x$-dependent phase of twist-2 gluon TMDs and twist-3 unpolarized gluon TMDs.

The Gist

Imagine you are trying to understand the internal structure of a super-dense, chaotic cloud of energy (like a proton or a heavy nucleus). To do this, physicists act like high-speed photographers, firing a tiny, high-energy particle (an electron) at the cloud. When the electron hits the cloud, it emits a burst of energy (a virtual photon) that smashes into the cloud's core, knocking out a pair of particles (a quark and an antiquark) that fly out in opposite directions like a pair of billiard balls.

This paper is about taking a very precise "snapshot" of that collision and figuring out exactly how the math changes when we stop making some of the "lazy" assumptions we usually make to keep the calculations simple.

Here is the breakdown using everyday analogies:

1. The "Shockwave" vs. The "Real Target"

The Old Way (The Eikonal Approximation):
Imagine the target cloud is a flat, frozen wall moving at the speed of light. When the photon hits it, the wall is so thin and fast that the photon just passes through it instantly, like a ghost walking through a sheet of paper. In physics, we call this the "shockwave approximation." It's a great shortcut, but it ignores the fact that the wall has thickness and that the particles inside it are actually moving and reacting.

The New Way (Finite-Energy Corrections):
This paper says, "Let's stop pretending the wall is infinitely thin." We need to account for the fact that the target has a little bit of thickness and that the collision takes a tiny, measurable amount of time.

• The Analogy: Imagine throwing a tennis ball at a moving train.
  • Old view: The train is a flat, invisible wall. The ball bounces off instantly.
  • New view: The train is a real object with windows, wheels, and a front car. The ball might hit the front, bounce off the side, or interact with the air pressure in front of the train. This paper calculates those extra, subtle interactions (called "next-to-eikonal" corrections) that happen because the target isn't just a flat wall.

2. The "Back-to-Back" Dance

The researchers are specifically looking at a scenario where the two particles fly out in perfect opposite directions (180 degrees apart).

• The Analogy: Think of a figure skater spinning and then suddenly throwing their arms out. If they throw them perfectly straight out, they are "back-to-back."
• In this specific dance, the particles are so perfectly aligned that we can use a special mathematical trick (the "correlation limit") to connect two different ways of describing the universe:
  1. CGC (Color Glass Condensate): A messy, chaotic, high-energy view of the target.
  2. TMDs (Transverse Momentum Dependent distributions): A cleaner, more organized view that maps out how the particles are moving inside the target.

3. The "Twist" in the Story

The paper's main discovery is about "Twist." In physics, "twist" is a fancy word for how complicated the internal structure of the target is.

• Twist-2 (The Simple View): This is the basic map of where the particles are and how fast they are moving. It's like looking at a city map and seeing the main highways.
• Twist-3 (The Complex View): This is the "fine print." It accounts for the subtle interactions, the "twists" in the road, and how the particles are actually jostling each other.

The Big Breakthrough:
The authors found that the "extra" math they had to do to account for the target's thickness (the finite-energy corrections) is exactly the same math needed to describe these complex "Twist-3" interactions.

• The Analogy: Imagine you are trying to predict the path of a leaf floating down a river.
  • Twist-2 is just knowing the speed of the current.
  • Twist-3 is knowing how the leaf spins and bumps into rocks.
  • The Paper's Discovery: They realized that if you stop pretending the river is a straight, smooth pipe (the "shockwave" assumption) and acknowledge the river has depth and currents (finite-energy), the math you use to describe the river's depth automatically gives you the math for how the leaf spins. You don't need two different sets of rules; they are the same rule written in two different languages.

4. Why Does This Matter?

This research is being prepared for the Electron Ion Collider (EIC), a massive new machine being built to take the ultimate "X-ray" of protons and nuclei.

• The EIC will be able to see these "Twist-3" effects and these "finite-energy" details.
• Before this paper, physicists had to guess how to connect the messy high-energy math with the clean TMD maps.
• Now, they have a bridge. They know exactly how to translate the messy, real-world collision data into a clean map of the gluon (the particle holding the nucleus together) distribution.

Summary

This paper is like a translator who realized that two different dialects of the same language are actually speaking about the same thing. By relaxing the "lazy" assumption that the target is a flat wall, the authors showed that the resulting complexity perfectly matches the complex "Twist-3" maps of the proton's interior. This ensures that when the new Electron Ion Collider starts taking pictures, we will know exactly how to interpret the blurry, complex details to see the clear picture of the universe's building blocks.

Technical Summary

Here is a detailed technical summary of the paper "Back-to-back dijet production in DIS with finite-energy corrections and twist-3 gluon TMDs" by Altinoluk et al.

1. Problem Statement and Motivation

The paper addresses the theoretical description of back-to-back dijet production in Deep Inelastic Scattering (DIS) off a dense gluon target (modeled by the Color Glass Condensate, CGC).

• Context: While standard high-energy calculations rely on the eikonal approximation (infinite energy limit where the target is treated as a shockwave), the upcoming Electron Ion Collider (EIC) will operate at energies where finite-energy effects become significant.
• The Gap: Existing calculations primarily focus on Next-to-Leading Order (NLO) corrections within the eikonal framework. However, there is a need to understand Next-to-Eikonal (NEik) corrections, which arise from relaxing the shockwave approximation. This involves:
  • Relaxing infinite Lorentz time dilation (introducing $z^-$ dependence).
  • Including subleading transverse components of the gauge field.
  • Relaxing the strict shockwave approximation to include target dynamics.
• Goal: To calculate the dijet cross-section at NEik accuracy in the "correlation limit" (back-to-back kinematics) and explicitly connect these finite-energy corrections to Twist-3 gluon Transverse Momentum Dependent (TMD) distributions, bridging the gap between the CGC and TMD factorization frameworks beyond leading twist.

2. Methodology

The authors employ the CGC effective field theory combined with a systematic expansion beyond the eikonal limit.

• Kinematics: The calculation focuses on the correlation limit where the produced jets are back-to-back.
  • Jet momenta: $k_1, k_2$.
  • Variables: Total momentum $P$ (relative transverse momentum between jets) and total momentum $k$ (small imbalance).
  • Limit: $|k| \ll |P|$ and dipole size $|r| \ll$ impact parameter $|b|$.
• Amplitude Expansion: The scattering amplitude is expanded in powers of $1/q^+$(where$q^+$ is the photon momentum).
  • Strict Eikonal: Neglects $z^-$ dependence of the background field.
  • Next-to-Eikonal (NEik): Includes three distinct contributions:
    1. Decorated Wilson Lines: Terms involving covariant derivatives on the Wilson lines (related to $F^-_i$).
    2. Transverse Field Strength: Terms involving the transverse field strength tensor $F_{ij}$.
    3. Target Dynamics: Terms arising from the $z^-$ dependence of the target field (related to $F^{+-}$).
• Cross-Section Calculation: The cross-section is computed by squaring the amplitude, keeping terms up to NEik order. This involves calculating the interference between the strict eikonal amplitude and the NEik correction amplitudes.
• Factorization: The resulting cross-section is expressed in terms of gluon field strength correlators ($\Phi^{\mu\nu;\rho\sigma}$). These correlators are then matched to the standard definitions of gluon TMDs to identify the twist structure.

3. Key Contributions

The paper makes several specific technical advances:

1. Explicit NEik Amplitudes: The authors derive the explicit forms of the NEik scattering amplitudes for a longitudinally polarized photon splitting into a quark-antiquark pair. They separate the contributions into:
   • $M^{(1)+(2)}$: From derivatives of Wilson lines ($F^-_i$).
   • $M^{(3)}$: From transverse field strength ($F_{ij}$).
   • $M^{\text{dyn. target}}$: From target dynamics ($F^{+-}$).
2. Factorization of Corrections: A crucial finding is the factorization of Next-to-Leading Power (NLP) kinematic corrections and Next-to-Eikonal operator corrections.
   • NLP corrections appear as kinematic factors modifying the hard scattering coefficient.
   • NEik corrections appear as modifications to the operator structure (involving light-cone time intervals $v'^+ - v^+$).
3. Twist-3 Identification: The work demonstrates how finite-energy corrections in the CGC formalism map directly to Twist-3 gluon TMDs. Specifically, the subeikonal terms correspond to unpolarized twist-3 distributions ($f^\perp_g$, $\bar{f}^\perp_g$, $g^\perp_g$, $\bar{g}^\perp_g$).
4. Vanishing Term: The authors show that the amplitude contribution involving the commutator $[\gamma^i, \gamma^j]$ (associated with $F_{ij}$) vanishes upon summation over helicities in the cross-section at this order, simplifying the final result.

4. Key Results

The final expression for the differential cross-section (Eq. 28) is written as a convolution of hard factors and TMDs:

$$\frac{d\sigma}{dP.S.} \propto \sum C_i(z_1, P, k) \times x \cdot \text{TMD}_i(x, k)$$

• Leading Twist (Twist-2): The dominant term involves the standard unpolarized gluon TMD $f_1^g(x, k)$. The NEik corrections introduce an $x$-dependent phase factor (resummed power corrections) to this term.
• Twist-3 Contributions: The NEik corrections generate terms proportional to:
  • $f^\perp_g$ and $\bar{f}^\perp_g$: Unpolarized twist-3 distributions related to the transverse momentum dependence and target dynamics.
  • $g^\perp_g$ and $\bar{g}^\perp_g$: Twist-3 distributions related to the transverse field strength.
• Hard Coefficients: The paper provides explicit analytical expressions for the hard coefficients ($C_{f_1^g}^L$, $C_{h_1^{\perp g}}^L$, etc.) which depend on the photon virtuality $Q$, the jet transverse momentum $P$, and the momentum fractions $z_1, z_2$.
• Structure of the Cross-Section: The result is organized into three parts based on field strength correlators:
  1. $F_\perp^- F_\perp^-$: Dominant, contains both twist-2 and twist-3 contributions.
  2. $F_\perp^- F_\perp^- F_\perp^-$: Triple field strength correlator.
  3. $F^{+-} F_\perp^-$: Mixed correlator representing target dynamics.

5. Significance

• Bridging CGC and TMD: This work provides a rigorous derivation showing that the CGC framework, when extended beyond the eikonal approximation, naturally reproduces the TMD factorization structure including Twist-3 effects. This validates the consistency of the two approaches.
• EIC Physics: The results are directly relevant for the Electron Ion Collider (EIC). As the EIC will probe the transition region between saturation (CGC) and dilute (TMD) regimes, understanding finite-energy (NEik) corrections is essential for precise extraction of gluon saturation scales and TMDs.
• Twist-3 Gluon Physics: The paper offers a concrete method to access unpolarized twist-3 gluon TMDs ($f^\perp_g$, etc.) through dijet production. These distributions are crucial for understanding the 3D structure of the nucleon and the role of gluon orbital angular momentum.
• Systematic Expansion: The work establishes a systematic procedure for expanding the CGC cross-section in powers of $1/Q$and$1/q^+$, separating kinematic power corrections from genuine dynamical (operator) corrections.

In summary, this paper is a foundational step in moving beyond the infinite-energy limit in high-energy QCD, providing the necessary theoretical tools to interpret future EIC data in terms of both saturation physics and higher-twist TMDs.