The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude

Abstract: As a test of the gluon scattering amplitude/Wilson loop duality, we evaluate the hexagonal light-like Wilson loop at two loops in N=4 super Yang-Mills theory. We compare its finite part to the Bern-Dixon-Smirnov (BDS) conjecture for the finite part of the six-gluon amplitude. We find that the two expressions have the same behavior in the collinear limit, but they differ by a non-trivial function of the three (dual) conformally invariant variables. This implies that either the BDS conjecture or the gluon amplitude/Wilson loop duality fails for the six-gluon amplitude, starting from two loops. Our results are in qualitative agreement with the analysis of Alday and Maldacena of scattering amplitudes with infinitely many external gluons.

• The paper tests the duality between the hexagon Wilson loop and the six-gluon amplitude, uncovering a non-trivial deviation from the BDS ansatz.
• It uses an explicit two-loop calculation in N=4 SYM to expose discrepancies in the expected iterative structure of MHV amplitudes.
• These findings suggest that either the BDS conjecture requires modification or the duality itself may fail at higher-loop levels, guiding future research.

The Hexagon Wilson Loop and the BDS Ansatz for the Six-Gluon Amplitude

The paper by Drummond, Henn, Korchemsky, and Sokatchev evaluates the hexagonal light-like Wilson loop at two loops in $\mathcal{N}=4$ super Yang-Mills theory and examines its relationship with the Bern-Dixon-Smirnov (BDS) conjecture for the six-gluon amplitude. This study is motivated by the conjectured duality between gluon scattering amplitudes and Wilson loops, which extends insights from the AdS/CFT correspondence into the weak coupling regime.

Main Findings

1. Gluon Amplitude/Wilson Loop Duality: The paper tests this duality by evaluating the hexagon Wilson loop and comparing its finite part with the BDS conjecture. It reveals that although both approaches align in their collinear limit, they differ by a non-trivial function of three dual conformally invariant variables.
2. Verification of the BDS Conjecture: Previously verified for four- and five-point amplitudes, the BDS ansatz suggests an iterative structure for MHV amplitudes in $\mathcal{N}=4$ SYM. However, this work finds a deviation from the BDS expectations for the six-gluon amplitude.
3. Two-Loop Calculation: The paper presents an explicit two-loop computation of the hexagon Wilson loop, revealing that the finite part $F_6^{\text{(WL)}}$ differs from the BDS counterpart $F_6^{\text{(BDS)}}$, implying a failure of either the BDS conjecture or the duality for the six-gluon amplitude.

Implications and Future Research

The findings have significant implications. Firstly, if the duality persists, the BDS conjecture must be modified to account for the discovered differences, implying additional terms that satisfy collinear behavior requirements. Conversely, if the BDS ansatz holds, the duality itself may be incorrect at higher loop levels for certain configurations.

Furthermore, the study suggests exploring the possible existence of functions of higher transcendentality. In this context, a function with transcendentality 4 beyond the BDS ansatz emerges as a candidate, potentially reshaping our understanding of the iterative structure in $\mathcal{N}=4$ SYM.

Conclusions

This research opens several avenues for further exploration. Critical among these is determining whether the discrepancies observed are an artifact specific to six-gluon amplitudes or indicative of a broader phenomenon at higher-point amplitudes. Additionally, the newly discovered function contained in the hexagon Wilson loop could provide insights into the mathematical structure underpinning scattering amplitudes in supersymmetric theories. Subsequent investigations could leverage numerical and analytical techniques to map out these discrepancies, verify dual conformal invariance, and explore new structures unaccounted for by existing conjectures.