Analytic Results for MHV Wilson Loops

TL;DR

The authors derive concise analytic two-loop Wilson-loop expressions for arbitrary n-point light-like polygons embedded in a 1+1 dimensional subspace of N=4 SYM, related to the AdS3 boundary. They show that in this kinematic sector all polylogarithmic content cancels, leaving a remainder function built purely from logarithms with a universal weight-four structure, and they provide explicit formulas for n=8,10,12 and a general n-point expression. The results are validated against multi-collinear limits and numerical evaluations, and agree with prior regular-polygon data, supporting a simple, universal logarithmic framework. The work suggests deeper structural simplifications at higher loops and invites exploration of alternative formalisms, including potential links to integrability observed at strong coupling.

Abstract

We obtain concise analytic formulae for Wilson loops computed on special n-point polygonal contours through two-loops in weakly coupled N=4 supersymmetric gauge theory. The contours we consider can be embedded into a (1+1)-dimensional subspace of the 4-dimensional gauge theory, corresponding to the boundary of the AdS_3 on the string theory side. Our analytic results hold for any number of edges, thus generalising to arbitrary n the recently derived expressions for 2-dimensional octagons. These polygonal Wilson loops have been conjectured to be equivalent to MHV scattering amplitudes in planar N=4 SYM.

Figures (3)

• Figure 1: The left figure shows the one-loop Wilson loop diagram which gives the finite part of the two-mass easy box function with massless momenta $p_i,p_j$ as in bht. On the right we represent the corresponding cross-ratio $u_{ij}$, the red dashed lines depicting the factors $x_{ij}^2$, $x_{i+1 j+1}^2$, $x_{i+1j}^2$, $x_{ij+1}^2$ in the definition of $u_{ij}$ in \ref{['uijdef']}. Later we will represent the cross-ratio as a single line stretched between edges $i$ and $j$ similarly to the gluon propagator.
• Figure 2: Figure illustrating equation (\ref{['ueq']}). On the left we represent in black the rectangular cross-ratio $(1- 1/u^\pm_{i\, j})$ and in red the rectangular cross-ratio $(1- 1/u^\pm_{i+1\, j+1}).$ On the right, on the other hand, in black we show the "crossed" cross-ratio $1-u_{i j+1}$ and in red $1-u^\pm_{i+1 j}$. Clearly the product on the left-hand side equals that on the right-hand side.
• Figure 3: The two loop $n$-point remainder function is given in \ref{['rn']}. It consists of a sum of terms of the form $\log ( u_{i_1 i_5}) \log ( u_{i_2 i_6}) \log ( u_{i_3 i_7}) \log ( u_{i_4 i_8})$ one term of which is represented pictorially here. One sums over all possible ways of drawing four mutually crossing lines connecting edges of the polygon, such that the parity of the lines alternates.