⚛️ nuclear theory

Gluonic nucleon energy correlators and fracture functions for Color Glass Condensate

This paper utilizes the Color Glass Condensate effective theory to demonstrate that gluonic nucleon energy correlators and fracture functions in the target fragmentation region are determined by the adjoint dipole $S$ -matrix, predicting a distinct $\cos 2\phi$ azimuthal asymmetry sensitive to the saturation scale and exhibiting significant nuclear suppression, thereby offering a novel probe for gluon saturation at the future Electron-Ion Collider.

Original authors: Heikki Mäntysaari, Yu Shi, Yossathorn Tawabutr, Xuan-Bo Tong

Published 2026-08-12

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CC BY 4.0

Original authors: Heikki Mäntysaari, Yu Shi, Yossathorn Tawabutr, Xuan-Bo Tong

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

Imagine the universe as a giant, cosmic Lego set. At the very bottom, the tiniest bricks are particles called quarks and gluons. These aren't just sitting still; they are zipping around at nearly the speed of light, glued together by a powerful force called the strong nuclear force to form protons and neutrons. For decades, scientists have been trying to take a "snapshot" of these bricks to see how they are arranged inside a proton. But there's a catch: when you zoom in really close, especially when the particles are moving very fast, they don't just act like individual bricks. They start to behave like a thick, sticky soup where the bricks are so crowded they merge into a single, dense state. This mysterious, crowded state is called the "Color Glass Condensate."

To understand this soup, scientists use a tool called Deep Inelastic Scattering (DIS). Think of this like firing a high-speed bullet (an electron) at a proton. Usually, scientists look at the debris flying away from the bullet's path to figure out what the proton was made of. But there is another part of the debris field: the stuff that gets kicked forward, in the same direction the proton was originally traveling. This is called the "Target Fragmentation Region." It's like looking at the dust cloud left behind after a car crash, rather than the pieces flying forward. This paper focuses on that specific dust cloud to find new clues about how the proton's internal "soup" behaves.

The researchers in this paper are investigating two specific tools to map this region: "fracture functions" and "energy correlators." You can think of a fracture function as a recipe that tells you the odds of finding a specific particle (like a gluon) inside the proton at the exact moment a specific piece of debris is spotted in the forward dust cloud. An "energy correlator" is a slightly more inclusive version; instead of looking for a specific particle, it just measures the total energy flowing in a specific direction in that forward cloud. The paper asks: What does this forward energy look like when the proton is made of this dense "Color Glass Condensate" soup?

The authors, using a sophisticated mathematical framework called the Color Glass Condensate effective theory, set out to calculate these energy patterns for gluons (the particles that carry the strong force). They found that in the high-speed limit they were studying, only two types of gluon patterns survive: the "unpolarized" ones (where the gluons are just randomly spinning) and the "linearly polarized" ones (where the gluons have a specific, aligned orientation, like a row of spinning tops all leaning the same way). Interestingly, they discovered that the complex, spinning-dependent patterns that exist in slower, less dense conditions vanish here. The entire forward energy pattern is determined by a single mathematical object called the "adjoint dipole S-matrix," which essentially measures how likely a gluon is to bounce off the dense soup without breaking it apart.

The most exciting finding comes from looking at the "linearly polarized" gluons. The team calculated that these aligned gluons create a very specific signature in the energy pattern: a wobble that repeats twice as the angle changes, known as a "cos2ϕ" asymmetry. Imagine the energy flowing out of the proton not as a perfect circle, but as a slightly squashed oval that rotates. The paper suggests that the size of this squashing is a direct measure of how "saturated" or crowded the gluon soup is.

To test this, the researchers ran numerical simulations for a future machine called the Electron-Ion Collider (EIC). They simulated collisions with both single protons and heavy gold nuclei (which are like giant, dense bags of protons and neutrons). Their results show a dramatic difference: while the energy pattern for a single proton shows a clear, strong "squashed oval" effect, the pattern for the heavy gold nucleus is significantly suppressed—it becomes much rounder, losing that distinct wobble. This suppression happens because the gold nucleus is so dense that the gluons are packed so tightly they screen each other out, a hallmark of the saturation effect.

The paper argues that this "squashed oval" effect is a much cleaner way to detect gluon saturation than previous methods. Older methods often required measuring two separate jets of particles, which can get messy with extra radiation and mathematical complications. This new method just needs to measure the total energy in one direction, making it theoretically simpler and experimentally easier to spot. The authors conclude that by measuring this specific energy wobble in the forward direction, the future Electron-Ion Collider will have a powerful, new window into the moment when gluons stop acting like individual particles and start behaving like a dense, saturated condensate.

Technical Summary: Gluonic Nucleon Energy Correlators and Fracture Functions for Color Glass Condensate

Problem Statement
Understanding the small- $x$ gluon structure of nucleons and nuclei is a central objective of high-energy Quantum Chromodynamics (QCD), particularly for the future Electron-Ion Collider (EIC). While the Color Glass Condensate (CGC) effective theory has successfully described transverse-momentum-dependent (TMD) parton distributions and observables in the current fragmentation region (CFR) of deep inelastic scattering (DIS), the target fragmentation region (TFR) remains less explored. In the TFR, observed particles originate from the target remnant, retaining correlations between the active struck parton and the final state that cannot be described by standard parton distribution functions (PDFs) or TMDs. These correlations are encoded in fracture functions and their inclusive counterparts, nucleon energy correlators (NECs).

Previous studies in the quark sector have established the connection between CGC Wilson-line correlators and quark NECs/fracture functions. However, the gluon sector presents specific challenges. A recent study on unpolarized gluon jet fracture functions suggested a surprising lack of sensitivity to gluon saturation effects, contrary to expectations given that the unpolarized gluon component is controlled by the adjoint dipole amplitude. Furthermore, the leading-twist components of gluonic fracture functions and NECs, particularly the linearly polarized gluon components and T-odd structures, had not been systematically calculated within the CGC framework starting from their operator definitions.

Methodology
The authors perform a systematic calculation of gluonic fracture functions and NECs within the CGC effective theory at eikonal accuracy (leading power in the high-energy limit). The methodology proceeds as follows:

Operator Definitions: The study begins with the rigorous operator definitions of gluonic NECs and fracture functions. NECs are defined via a correlation matrix involving the gluon field strength tensor and a light-like gauge link (Wilson line) in the adjoint representation, measuring the energy flow from target remnants. Fracture functions are defined similarly but involve the measurement of a specific hadron.
CGC Formalism and Shockwave Approximation: The calculations are performed in the shockwave formalism, where the fast-moving target is treated as a dense source of small- $x$ gluons. The interaction is encoded in an instantaneous shockwave represented by Wilson lines. The authors work in the light-cone gauge ( $A^- = 0$ ) and consider the eikonal limit where the target spin effects are subleading.
Amplitude-Level Calculations: The fracture-function amplitudes are calculated at the amplitude level by considering two distinct diagrams: (a) the real gluon emitted before the shockwave interaction, and (b) the real gluon emitted after the shockwave interaction (attaching to the gauge link).
Squaring and Interference: The correlation matrices for fracture functions are obtained by squaring the amplitudes, resulting in three contributions: the square of diagram (a), the square of diagram (b), and the interference term between them.
Projection and Sum Rules: The authors project out the independent components of the correlation matrix to isolate specific gluonic fracture functions. They then utilize the inclusive energy sum rule to derive the corresponding NECs from the fracture functions, ensuring a one-to-one correspondence between the two sets of observables.
Numerical Implementation: For phenomenological predictions, the authors employ the large- $N_c$ limit, expressing the adjoint dipole S-matrix as the square of the fundamental quark dipole S-matrix. They solve the running coupling Balitsky-Kovchegov (rcBK) evolution equation with a modified McLerran-Venugopalan initial condition to model the small- $x$ evolution of the dipole amplitudes for both proton and gold nucleus targets.

Key Contributions and Results

Selection of Non-Vanishing Components: The authors find that at eikonal accuracy, only two of the eight independent leading-twist gluonic components survive for an unpolarized target: the unpolarized gluon component ( $f_1^g$ ) and the linearly polarized gluon component ( $h_{1}^{T,g}$ ). All target-spin-dependent components (including T-odd structures) vanish in the strict eikonal limit. This is attributed to the fact that the eikonal gluon-dipole S-matrix is real and C-even, precluding contributions from the spin-dependent odderon which is purely imaginary.
Adjoint Dipole Dependence: Both the unpolarized and linearly polarized gluon fracture functions and NECs are determined entirely by the adjoint dipole S-matrix.
Dilute Limit and Collinear Matching: In the dilute limit (large transverse momentum or large angle), the calculated CGC results match the standard collinear factorization results. The unpolarized gluon fracture function exhibits a $1/p_{\perp}^2$ scaling, while the linearly polarized component is additionally suppressed by a kinematic factor of $M^2/p_{\perp}^2$ .
Saturation Sensitivity of Linearly Polarized Gluons: A central finding is the behavior of the linearly polarized gluon NEC ( $h_{1}^{T,g}$ ). Unlike the unpolarized gluon NEC, which shows negligible nuclear suppression, the linearly polarized component exhibits substantial nuclear suppression in the small-angle region ( $\theta Q \lesssim Q_s$ ).
Cosine-Modulated Asymmetry: The linearly polarized gluon NEC generates a characteristic $\cos 2\phi$ azimuthal asymmetry in the DIS energy pattern in the TFR. The authors derive that at leading order in $\alpha_s$ and small $x_B$ , this asymmetry is governed by the ratio of the linearly polarized gluon NEC ( $h_{1}^{T,g}$ ) to the unpolarized quark NEC ( $f_1^q$ ).
Numerical Predictions for EIC: Numerical analysis for EIC kinematics ( $x_B = 5 \times 10^{-3}$ , $Q^2 = 25 \text{ GeV}^2$ ) predicts a significant nuclear suppression of the $\langle \cos 2\phi \rangle$ asymmetry in $e+A$ collisions compared to $e+p$ collisions. The nuclear modification factor is well below unity, particularly at small angles, providing a clear signature of gluon saturation.

Significance and Claims
The paper claims to provide a unified framework for probing small- $x$ nucleon tomography in the TFR at eikonal accuracy, extending previous quark-sector results to the gluon sector. The primary significance lies in identifying the linearly polarized gluon NEC as a novel and sensitive probe of gluon saturation.

The authors argue that the proposed observable—the $\cos 2\phi$ asymmetry in the TFR energy pattern—offers distinct theoretical and experimental advantages over traditional TMD-based dijet observables:

Theoretical Cleanliness: Formulated within collinear factorization, the observable is free from Sudakov suppression and contamination from soft-gluon radiation that typically affects TMD-based dijet correlations.
Experimental Feasibility: It requires only the inclusive measurement of a single energy flow (calorimetric measurement) rather than the complex reconstruction of a dijet system.
Saturation Probe: The asymmetry's dependence on the ratio of adjoint to fundamental dipole amplitudes makes it highly sensitive to the saturation scale. The predicted substantial nuclear suppression in $e+A$ collisions serves as a distinctive signature for the onset of gluon saturation at the future EIC.

The authors conclude that while their work focuses on the eikonal limit, future extensions could include subeikonal corrections to access spin-dependent and C-odd structures, as well as investigations into process dependence and universality in hadronic collisions.

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