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

TL;DR

The paper tests the gluon scattering amplitude/Wilson loop duality in N=4 SYM by computing the two-loop hexagon (six-point) light-like Wilson loop and comparing its finite part to the BDS finite part for six-gluon amplitudes. It shows that F6_WL differs from F6_BDS by a non-trivial, symmetric function f(u1,u2,u3) of the three dual conformal cross-ratios, though the collinear limit behavior agrees with the amplitude analysis, suggesting a breakdown of either the BDS ansatz or the duality at two loops. The authors perform a detailed two-loop calculation using the maximally non-Abelian color factor and confirm the cross-ratio dependence via numerical evaluation, aligning with qualitative expectations from Alday-Maldacena for large n. They discuss three possible interpretations and emphasize the need for direct two-loop six-gluon amplitude results, noting that the corrective function has transcendentality four.

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.

Figures (3)

• Figure 1: 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 2: The $\gamma-$dependence of the function $\widehat{f}^{(2)}(\gamma,u,1-u)$, Eq. (\ref{['sub']}), for different values of the parameter $u=0.5$ (lower curve), $u=0.3$ (middle curve) and $u=0.1$ (upper curve).
• Figure 3: The $u-$dependence of the function $\widehat{f}^{(2)}(\gamma,u,1-u)$, Eq. (\ref{['sub']}), for different values of the parameter $\gamma=0.001$ (lower curve), $\gamma=0.01$ (middle curve) and $\gamma=0.1$ (upper curve).