⚛️ high-energy theory

Three-Reggeon exchange in $\mathcal{N}=4$ SYM to leading logarithmic accuracy

This paper computes leading and next-to-leading logarithmic contributions to eight-point amplitudes in planar $\mathcal{N}=4$ Super Yang-Mills theory by employing an effective field theory approach to evaluate four-loop three-Reggeon exchange diagrams and proposing a compact Fourier-Mellin representation for two-Reggeon exchange, thereby providing novel high-loop results for octagon amplitudes in multi-Regge kinematics.

Original authors: Claude Duhr, Zhenjie Li, Chi Zhang

Published 2026-08-26

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CC BY 4.0

Original authors: Claude Duhr, Zhenjie Li, Chi Zhang

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

In the realm of theoretical physics, there is a specific theory called Super Yang-Mills that serves as a perfect laboratory for scientists. It is a simplified version of the rules that govern how particles interact, stripped of the messy complications found in the real world, yet it retains the deep mathematical structures that make the universe tick. Within this theory, physicists study scattering amplitudes, which are essentially calculations that predict the likelihood of particles colliding and scattering in specific ways. For decades, researchers have been trying to map out these probabilities with extreme precision, hoping to uncover hidden symmetries that might explain the fundamental nature of reality. A key challenge in this pursuit is understanding what happens when particles are produced at very high energies and travel in a specific, ordered sequence. In this scenario, known as multi-Regge kinematics, the particles are arranged like beads on a string, with their speeds and directions following a strict hierarchy. For a long time, scientists believed they had a complete picture of how these interactions worked in certain energy zones, but as they looked at more complex collisions involving eight or more particles, cracks began to appear in their understanding.

The core of the mystery lies in how these particles exchange energy and momentum. In the simplified view, this exchange is mediated by objects called Reggeons, which act like carriers of force between the colliding particles. For simpler collisions, the interaction is dominated by the exchange of just two of these carriers. However, when the collision involves eight particles arranged in a specific "zigzag" pattern of energy flow, a new and more complicated phenomenon emerges: the exchange of three Reggeons simultaneously. Until now, no one had successfully calculated the precise outcome of this three-way exchange, leaving a gap in the theoretical map of high-energy physics. This gap was significant because it meant that for certain complex collisions, the existing mathematical tools could not predict the final state of the particles with the required accuracy.

A team of researchers has now taken the first major step toward filling this gap. They focused on the eight-particle collision in that specific zigzag energy configuration, a scenario where both two-Reggeon and three-Reggeon exchanges contribute to the final result. To solve this, they employed a specialized approach known as effective field theory. Think of this method as a way to zoom in on the most important interactions while temporarily ignoring the less relevant details, allowing the complex web of particle exchanges to be broken down into a set of manageable rules. Using these rules, the team constructed a new set of instructions, similar to a recipe for a complex dish, that allowed them to calculate the contribution of the three-Reggeon exchange directly. They performed these calculations up to four loops, a term that refers to the level of precision and complexity in the mathematical steps, effectively pushing the boundaries of what has been computed before.

The researchers did not stop at just calculating the new three-Reggeon piece. They also realized that to get the full picture, they needed to combine this new result with the known contributions from the two-Reggeon exchange. They proposed a compact and elegant mathematical framework to describe the two-Reggeon part in this specific energy zone, ensuring that it matched all the known behaviors of the theory. By weaving together the new three-Reggeon calculation with this refined two-Reggeon description, they were able to produce a complete and consistent prediction for the eight-particle collision. They tested their work by checking if it matched existing data for simpler cases, and it did, confirming that their new approach was sound.

The results of this study are novel and extend the frontiers of knowledge. The team provided the first-ever calculation for the eight-particle collision in this specific energy region up to four loops of precision for the most common particle configuration, and up to three loops for more complex configurations. Furthermore, they showed that their method is not limited to just eight particles; they derived a general formula that describes the three-Reggeon contribution for any number of particles in a similar collision setup. This work suggests that the complex interactions of three force carriers can be understood and calculated systematically, opening the door to predicting the outcomes of even more intricate particle collisions. While the calculations were performed within the simplified Super Yang-Mills theory, the techniques and insights gained could eventually help physicists understand similar phenomena in the real world, such as the behavior of high-energy cosmic rays or the elusive Odderon, a rare state of matter predicted in the theory of strong nuclear forces. The paper stands as a proof of concept that these highly complex, multi-particle exchanges can be tamed and understood, providing a solid foundation for future explorations into the deepest layers of particle physics.

Technical Summary: Three-Reggeon Exchange in N = 4 SYM to Leading Logarithmic Accuracy

Problem Statement
The scattering amplitudes in planar $\mathcal{N}=4$ Super Yang-Mills (SYM) theory exhibit remarkable symmetries, including dual conformal invariance and Yangian symmetry. While the BDS ansatz successfully describes the infrared structure of these amplitudes, it is incomplete starting from six external particles, requiring a dual-conformally invariant remainder function. In the multi-Regge kinematics (MRK), where produced particles are strongly ordered in rapidity, the BDS ansatz is valid in the Euclidean region but develops unphysical analytic structures (violating Steinmann conditions) when analytically continued to specific Mandelstam regions.

For Mandelstam regions where all centrally produced particles have negative energy (denoted as $[k, l]$ ), the amplitude is governed by two-Reggeon exchange and can be described via the Balitsky-Fadin-Kuraev-Lipatov (BFKL) effective field theory (EFT) and integrability. However, for eight or more particles, new Mandelstam regions exist (denoted as $\{k, l\}$ ) where not all central particles have negative energy. In these regions, the amplitude receives contributions from the exchange of three (or more) Reggeons. Prior to this work, no analytic results existed for scattering amplitudes in MRK within these specific Mandelstam regions where three-Reggeon exchange yields a non-trivial contribution.

Methodology
The authors employ an effective field theory approach to compute the leading logarithmic accuracy (LLA) contributions from three-Reggeon exchange. The methodology involves:

BFKL Effective Field Theory: The authors utilize a two-dimensional EFT formulation where $s$ -channel cuts are treated via on-shell conditions, and $t$ -channel exchanges are described by Reggeons. They apply specific Feynman rules involving Lipatov vertices, peripheral vertices, and BFKL bridges in the transverse plane.
Leading Logarithmic Accuracy (LLA) Computation: They derive a set of Feynman rules allowing for the explicit evaluation of diagrams contributing to three-Reggeon exchange at LLA. This involves calculating loop integrals that result in single-valued polylogarithms. Infrared divergences arising in these integrals are handled via a specific subtraction scheme involving triangle integrals.
Fourier-Mellin Representation for Two-Reggeon Exchange: To obtain the full amplitude at next-to-leading-logarithmic accuracy (NLLA), the three-Reggeon contribution (which enters at NLLA for the full amplitude) must be combined with the two-Reggeon exchange contribution. The authors propose a compact Fourier-Mellin representation for the two-Reggeon exchange in the $\{k, l\}$ regions. This representation utilizes the same impact factors and central emission vertices known from $[k, l]$ regions but incorporates modified integration contours and phase factors to account for the specific kinematic region.
Validation: The proposed representation for the two-Reggeon exchange is validated by demonstrating that it reproduces the known multi-Regge limits of the eight-point MHV amplitude up to three loops.

Key Contributions and Results

Three-Reggeon Exchange at LLA: The paper provides the first explicit evaluation of the three-Reggeon exchange contribution for the eight-point amplitude in the "zigzag" region (a specific Mandelstam region denoted as $\{4, 7\}$ ) up to four loops. The authors derive a simple set of Feynman rules that allow these contributions to be computed in terms of single-valued polylogarithms.
Compact Fourier-Mellin Representation: A novel, compact Fourier-Mellin integral representation is proposed for the two-Reggeon exchange contribution in the $\{k, l\}$ regions. This formula is valid to all orders in perturbation theory and relies on ingredients (impact factors, BFKL eigenvalues, central vertices) conjectured to be known to all orders via integrability.
Novel Amplitude Results:
- Four-Loop MHV: The authors combine the new three-Reggeon results with the proposed two-Reggeon representation to obtain the complete result for the eight-point MHV amplitude in the zigzag region up to four loops at next-to-leading-logarithmic accuracy (NLLA).
- Three-Loop Non-MHV: Novel results are presented for non-MHV (NMHV) amplitudes in the same region up to three loops.
- Arbitrary Multiplicity: The paper presents a compact result for the three-Reggeon contribution to all three-loop MHV amplitudes with an arbitrary number of legs in the relevant Mandelstam region. The authors observe that the LLA three-loop three-Reggeon contribution for any $N$ -point amplitude is entirely determined by the eight-point result, analogous to the factorization observed in two-Reggeon exchange.
Analytic Continuation: The work includes a detailed procedure for the analytic continuation of known octagon results to the zigzag region, confirming consistency with existing perturbative data at two and three loops.

Significance
The paper claims to take the "first steps toward analytic results for scattering amplitudes in MRK in Mandelstam regions that receive contributions from three-Reggeon exchange." The significance of the work lies in:

Closing a Gap: It addresses the lack of understanding regarding amplitudes in Mandelstam regions where $m \geq 3$ Reggeon exchanges are non-trivial.
Methodological Advancement: It demonstrates that the BFKL-EFT approach can be successfully extended to compute multi-Reggeon exchanges beyond the two-Reggeon case, providing a practical tool for evaluating higher-loop amplitudes.
Integrability Connection: The results suggest that the building blocks for three-Reggeon exchange may also be accessible via integrability, similar to the two-Reggeon case. The authors note that the three-Reggeon evolution corresponds to a non-compact $SL(2, \mathbb{C})$ spin chain, representing a genuinely non-trivial instance of this structure.
Future Applications: The results provide new boundary data for bootstrap programs aiming to determine higher-point amplitudes in full kinematics. Additionally, the authors suggest that the methods developed could offer insights into the Odderon in QCD, although they acknowledge the differences in color representation and dynamical settings.

The paper remains modest in its scope, focusing specifically on leading logarithmic accuracy for three-Reggeon exchange and next-to-leading-logarithmic accuracy for the combined amplitude, without claiming to solve the full resummation of three-Reggeon exchange to all orders or to fully resolve the integrability of the associated spin chain beyond the LLA level.

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