Are scattering amplitudes dual to super Wilson loops?

TL;DR

Belitsky, Korchemsky, and Sokatchev analyze two Minkowski-space supersymmetric Wilson loop proposals as potential duals to non-MHV amplitudes in planar \\mathcal{N}=4 SYM. They derive supersymmetric Ward identities and compute one-loop anomalies arising from light-cone singularities, showing that chiral Q-supersymmetry is broken at this order and that the Wilson-loop constructions fail to match non-MHV amplitudes. By solving the anomalous Ward identities, they obtain the explicit one-loop form of the rectangular Wilson loop and verify a conformal anomaly, confirming the breakdown of the duality beyond MHV. The work also discusses implications for alternative dual descriptions and the role of on-shell vs off-shell formulations in highly supersymmetric theories.

Abstract

The MHV scattering amplitudes in planar N=4 SYM are dual to bosonic light-like Wilson loops. We explore various proposals for extending this duality to generic non-MHV amplitudes. The corresponding dual object should have the same symmetries as the scattering amplitudes and be invariant to all loops under the chiral half of the N=4 superconformal symmetry. We analyze the recently introduced supersymmetric extensions of the light-like Wilson loop (formulated in Minkowski space-time) and demonstrate that they have the required symmetry properties at the classical level only, up to terms proportional to field equations of motion. At the quantum level, due to the specific light-cone singularities of the Wilson loop, the equations of motion produce a nontrivial finite contribution which breaks some of the classical symmetries. As a result, the quantum corrections violate the chiral supersymmetry already at one loop, thus invalidating the conjectured duality between Wilson loops and non-MHV scattering amplitudes. We compute the corresponding anomaly to one loop and solve the supersymmetric Ward identity to find the complete expression for the rectangular Wilson loop at leading order in the coupling constant. We also demonstrate that this result is consistent with conformal Ward identities by independently evaluating corresponding one-loop conformal anomaly.