Pulling the straps of polygons

TL;DR

The paper develops an OPE-based bootstrap for null polygon Wilson loops in N=4 SYM, showing that two-loop discontinuities are determined by the one-loop seed via an anomalous-dimension kernel and the symbol of polylogarithms. It provides a detailed hexagon analysis, decomposing the one-loop source into conformal blocks and reconstructing the two-loop discontinuity D_2 through a symbolic approach, then reassembling the full two-loop remainder function. The results support the conjecture that OPE discontinuities suffice to fix the full two-loop answer for arbitrary polygon sides and highlight the power of momentum-twistor and flux-tube formalisms, with broader implications for integrability and higher-point amplitudes.

Abstract

Using the Operator Product Expansion for Wilson loops we derive a simple formula giving the discontinuities of the two loop result in terms of the one loop answer. We also argue that the knowledge of these discontinuities should be enough to fix the full two loop answer, for a general number of sides. We work this out explicitly for the case of the hexagon and rederive the known result.

Figures (4)

• Figure 1: Correlation function between two Polygonal Wilson loops, which is given by a single gluon exchange. In (a) we consider a Polygon where all the points in the $X$ contour are spacelike separated from all points in the $Y$ contour. (b) We take a limit of (a) where a segment of one contour $(Y_k,Y_{k+1})$ is on the same null line as a cusp of the other contour $X_i$. The rest of the points are spacelike separated.
• Figure 2: Triangulation of a null polygon Wilson loops into the two type of triangles. The two types of triangles correspond to the ones bounded by $\lambda$'s which are all different or the ones with the same $\lambda$ for all three edges.
• Figure 3: (a) We see the hexagon. The sides 1 and 4 are the left and right sides. The vertices 1 and 4 coincide with the reference square. In (b) we see the contours that give rise to the U(1) source $r_{U(1)}$.
• Figure 4: A polygon in a mixed kinematics called Hybrid. In the OPE limit $u\to\infty$ the two loops remainder function can be divided into two pieces as $R_2=t\,D+\widetilde{D}$. The function $D(s,t)$ is computed in this section.