On planar gluon amplitudes/Wilson loops duality

TL;DR

This paper provides a two-loop calculation confirming the duality between planar gluon scattering amplitudes and Wilson loops in N=4 SYM theory, extending the evidence beyond one loop and supporting the BDS conjecture.

Contribution

It explicitly verifies the gluon amplitude/Wilson loop duality at two loops and proposes an anomalous conformal Ward identity that constrains the finite part of the Wilson loop.

Findings

Duality holds at two loops for four-gluon amplitudes.

Anomalous conformal Ward identity constrains the finite part.

Results agree with the BDS conjecture for multi-gluon amplitudes.

Abstract

There is growing evidence that on-shell gluon scattering amplitudes in planar N=4 SYM theory are equivalent to Wilson loops evaluated over contours consisting of straight, light-like segments defined by the momenta of the external gluons. This equivalence was first suggested at strong coupling using the AdS/CFT correspondence and has since been verified at weak coupling to one loop in perturbation theory. Here we perform an explicit two-loop calculation of the Wilson loop dual to the four-gluon scattering amplitude and demonstrate that the relation holds beyond one loop. We also propose an anomalous conformal Ward identity which uniquely fixes the form of the finite part (up to an additive constant) of the Wilson loop dual to four- and five-gluon amplitudes, in complete agreement with the BDS conjecture for the multi-gluon MHV amplitudes.