The Two-Loop Hexagon Wilson Loop in N = 4 SYM

TL;DR

The paper delivers the first complete analytic computation of the two-loop remainder function for the hexagon Wilson loop in planar N=4 SYM. By exploiting Regge exactness in quasi-multi-Regge kinematics, Mellin-Barnes representations, and the Goncharov polylogarithm framework, it expresses the remainder R_{6,WL}^{(2)}(u1,u2,u3) as a weight-four function of conformal cross ratios. It provides a detailed treatment of the hardest integral, presents the full analytic form (and a companion electronic version) in terms of Goncharov polylogarithms, and analyzes asymptotic and equal-cross-ratio limits, including exact values at special points. These results reinforce the Wilson loop–amplitude duality and furnish a robust, scalable approach for higher-point polygons and more complex kinematics.

Abstract

In the planar N=4 supersymmetric Yang-Mills theory, the conformal symmetry constrains multi-loop n-edged Wilson loops to be given in terms of the one-loop n-edged Wilson loop, augmented, for n greater than 6, by a function of conformally invariant cross ratios. That function is termed the remainder function. In a recent paper, we have displayed the first analytic computation of the two-loop six-edged Wilson loop, and thus of the corresponding remainder function. Although the calculation was performed in the quasi-multi-Regge kinematics of a pair along the ladder, the Regge exactness of the six-edged Wilson loop in those kinematics entails that the result is the same as in general kinematics. We show in detail how the most difficult of the integrals is computed, which contribute to the six-edged Wilson loop. Finally, the remainder function is given as a function of uniform transcendental weight four in terms of Goncharov polylogarithms. We consider also some asymptotic values of the remainder function, and the value when all the cross ratios are equal.