Two-Loop Polygon Wilson Loops in N=4 SYM

TL;DR

This work computes two-loop corrections to arbitrary $n$-gon lightlike Wilson loops in ${\cal N}=4$ SYM, uncovering a remainder function beyond the ABDK/BDS ansatz that depends only on dual conformal cross-ratios. By combining non-abelian exponentiation with a fixed set of master integrals, the authors numerically evaluate all relevant two-loop diagrams for $n=6,7,8$ and verify dual conformal invariance and correct collinear behavior against the corresponding amplitudes. They demonstrate that the six-, seven-, and eight-point Wilson loops exhibit the expected cross-ratio dependence and collinear factorization, providing strong evidence for the Wilson loop–amplitude duality at two loops and suggesting a path to determine all planar $n$-point two-loop MHV amplitudes in the theory. The methodology relies on a stable numerical framework for evaluating dimensionally regulated Feynman integrals and a rigorous analysis of the remainder function's kinematic structure, including its invariance properties and limiting behaviors.

Abstract

We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in N=4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n=6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n=7,8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of "master integrals", which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the N=4 theory.

Figures (13)

• Figure 1: On the left we represent the one-loop Wilson loop diagram which gives the finite part of the two-mass easy box function with massless momenta $p_i,p_j$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{['ourcrossratios']}.
• Figure 2: The hard diagram.
• Figure 3: The curtain diagram.
• Figure 4: The cross diagram.
• Figure 5: The Y diagram together with the self-energy diagram.