Hexagon Wilson loop = six-gluon MHV amplitude

TL;DR

This paper provides strong two-loop evidence for the Wilson loop/ MHV amplitude duality in planar N=4 SYM by comparing the finite part of the light-like hexagon Wilson loop with the two-loop finite part of the six-gluon MHV amplitude. It confirms that both objects deviate from the BDS ansatz by the same nontrivial remainder function dependent on three dual conformal cross-ratios, thereby supporting the duality to all orders in the coupling for arbitrary n. The analysis combines a detailed two-loop Wilson-loop calculation (utilizing non-Abelian exponentiation and subtraction methods) with numerical six-gluon amplitude results, demonstrating consistency across dual descriptions and underscoring the breakdown of BDS at six points. The work also highlights the potential role of dual conformal symmetry and deeper structures (e.g., integrability) in organizing planar amplitudes and Wilson loops.

Abstract

We compare the two-loop corrections to the finite part of the light-like hexagon Wilson loop with the recent numerical results for the finite part of the MHV six-gluon amplitude in N=4 SYM theory by Bern, Dixon, Kosower, Roiban, Spradlin, Vergu and Volovich (arXiv:0803.1465 [hep-th]) and demonstrate that they coincide within the error bars and, at the same time, they differ from the BDS ansatz by a non-trivial function of (dual) conformal kinematical invariants. This provides strong evidence that the Wilson loop/scattering amplitude duality holds in planar N=4 SYM theory to all loops for an arbitrary number of external particles.

Figures (3)

• Figure 1: The conjectured duality relation between the gluon scattering amplitude $\mathcal{M}_n$ and the Wilson loop $W(C_n)$. The dashed lines depict gluons and the double lines the integration contour $C_6$. The momenta of the incoming gluons are identified as the light-like segments of the integration contour, $p_i \sim x_i - x_{i+1}$.
• Figure 2: The maximally non-Abelian Feynman diagrams of different topology contributing to $F_6^{\rm (WL)}$. The double lines depict the integration contour $C_6$, the dashed lines the gluon propagator and the blob the one-loop polarization operator.
• Figure 3: The auxiliary Feynman diagrams defined in (\ref{['I-aux']}). The double line depicts the integration contour $C_6$, the dashed line the gluon propagator and the box the fictitious three-gluon vertex (\ref{['J-integral']}).