Bootstrapping the three-loop hexagon (1108.4461v2)

Abstract: We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar N=4 super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol's entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3-->3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

• The paper introduces a novel symbolic bootstrap method to determine the three-loop hexagon remainder function using OPE constraints and an ansatz for the symbol entries.
• It validates BFKL predictions in the multi-Regge limit, highlighting the consistency between Wilson loops and MHV scattering amplitudes under dual conformal symmetry.
• The study offers practical insights for simplifying high-loop amplitude calculations in gauge theories, paving the way for exploring more complex kinematic configurations.

Overview of "Bootstrapping the three-loop hexagon"

The paper, "Bootstrapping the three-loop hexagon," authored by Lance J. Dixon, James M. Drummond, and Johannes M. Henn, explores the core aspects of scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills (SYM) theory, focusing particularly on the hexagon Wilson loop configuration. The analysis applies constraints from the Operator Product Expansion (OPE) in the near-collinear limit to uncover the intricate structure of the remainder function at three-loop order.

Summary

The research investigates the interplay between Wilson loops and maximally-helicity-violating (MHV) scattering amplitudes, emphasizing their equivalence under dual conformal symmetry in $\mathcal{N}=4$ SYM. Motivated by the duality observed between amplitudes and Wilson loops, the paper explores the implications of this symmetry at a high-loop level, specifically targeting three-loop hexagon configurations.

Initially, the paper scrutinizes the hexagonal Wilson loop dual to the six-point MHV amplitude using constraints derived from the OPE expansion, leveraging this framework to construct the symbol of the remainder function at three loops. By assuming a plausible ansatz for the symbol's entries in the function, the authors are able to constrain the symbol of the remainder function up to two constants which are undetermined by present methods.

A significant achievement of this paper is the validation of the BFKL (Balitsky–Fadin–Kuraev–Lipatov) prediction within the leading-logarithm approximation in the multi-Regge limit, confirming the robustness of their ansatz and consistency in the duality between Wilson loops and MHV amplitudes. The symbolic expression developed in this paper provides a strong consistency check against BFKL predictions, while also anticipating the form of the full three-loop remainder function in the multi-Regge limit, extending beyond the leading-logarithm approximation.

Implications

The paper provides a deeper insight into understanding the analytic structure of scattering amplitudes in $\mathcal{N}=4$ SYM. A key contribution of this research is the elucidation of the three-loop remainder function using symbolic methods, enhancing the comprehension of dual conformal symmetry in high-energy amplitudes.

On a more practical level, these findings have implications for constructing amplitudes in gauge theories more broadly, providing a potential template for approaching problems of analytic complexity found at higher loops. The symbolic methods employed in the paper are anticipated to influence future computational strategies in the paper and simplification of amplitude calculations in field theories.

Future Directions

The research opens several avenues for future exploration. The authors anticipate further investigation into determining the undetermined constants by leveraging additional constraints, or perhaps through numerical evaluations of amplitudes themselves. Moreover, there is scope for expansion beyond six-point kinematics to more complex scenarios that could involve higher numbers of points or different kinematic configurations.

Understanding the broader applicability of these methods across different field theories, especially where dual conformal properties play a significant role, could pave the way to potent new methods for analyzing amplitudes in field theories beyond $\mathcal{N}=4$ SYM.

In conclusion, "Bootstrapping the three-loop hexagon" provides a detailed and profound exploration of three-loop amplitudes using symbolic methods, validating theoretical predictions and offering a structured pathway for future research to further probe the mysteries of gauge theory dynamics under duality conditions.