Simplicity of Polygon Wilson Loops in N=4 SYM

TL;DR

This work investigates whether polygonal Wilson loops in N=4 SYM exhibit a universal remainder function that matches weak- and strong-coupling regimes. By computing the two-loop octagon remainder and analyzing regular 2n-gons, the authors find a linear relation ${\cal R}_{2n}^{(L)} = A^{(L)} {\cal R}_{2n}^{\rm strong}$ with a coupling-dependent factor $A^{(L)}$ and, in the octagon case, an additive offset ${B}^{(L)}$ that can be absorbed into the strong-coupling baseline. The results suggest a universal, n- and kinematics-independent mapping between weak- and strong-coupling remainder functions, motivating further analytic studies. If confirmed, this universality would illuminate the dual conformal structure of MHV amplitudes and simplify their all-order behavior across coupling regimes.

Abstract

Wilson loops with lightlike polygonal contours have been conjectured to be equivalent to MHV scattering amplitudes in N=4 super Yang-Mills. We compute such Wilson loops for special polygonal contours at two loops in perturbation theory. Specifically, we concentrate on the remainder function R, obtained by subtracting the known ABDK/BDS ansatz from the Wilson loop. First, we consider a particular two-dimensional eight-point kinematics studied at strong coupling by Alday and Maldacena. We find numerical evidence that R is the same at weak and at strong coupling, up to an overall, coupling-dependent constant. This suggests a universality of the remainder function at strong and weak coupling for generic null polygonal Wilson loops, and therefore for arbitrary MHV amplitudes in N=4 super Yang-Mills. We analyse the consequences of this statement. We further consider regular n-gons, and find that the remainder function is linear in n at large n through numerical computations performed up to n=30. This reproduces a general feature of the corresponding strong-coupling result.

Figures (10)

• Figure 1: Alternative kinematics, conformally equivalent to \ref{['8ptkinematics']}, in which the spatial projection of the polygon circumscribes the unit circle.
• Figure 2: The octagon remainder function in the special kinematics considered in amoctopus as a function of ${\rm Re}(m)$ and ${\rm Im}(m)$. The figure on the left represents the strong-coupling result ${\cal R}_{8}^{\rm AM}$ derived in \ref{['AMR8']}, while on the right we plot our two-loop result ${\cal R}_{8}^{\rm (2)}$.
• Figure 3: The superposition of the 3D plots for $\overline{{{\cal R}}}_8^{\,\rm AM}(m)$ and $\overline{{{\cal R}}}_8^{(2)}(m)$.
• Figure 4: A "bird's eye view" of Figure \ref{['ramagainst2looprescaled']} computed in kinematics A.
• Figure 5: This graph is a plot of $\overline{{{\cal R}}}_8^{\,\rm AM}(m)$ and $\overline{{{\cal R}}}_8^{(2)}(m)$ as a function of $|m|$ for $\phi=\pi/4$ computed in kinematics C.