Multichannel Conformal Blocks for Polygon Wilson Loops

TL;DR

Sever and Vieira develop a multichannel conformal block framework for null polygon Wilson loops that generalizes the OPE analysis beyond six edges. They explicitly construct heptagon blocks using two commuting SL(2) Casimirs, decompose the one-loop result, and predict the two-loop OPE discontinuity by dressing flux-tube excitations, with the promise that symbol-level data determine the full two-loop answer. They then generalize the method to polygons with arbitrary n by introducing multiple OPE channels and reference squares, yielding a scalable, multi-variable block structure (Appell-type in many variables) and practical building-block decompositions for higher-loop predictions. The work offers a concrete route to extract OPE data and cross-check against symbol-based amplitude results, enhancing understanding of the analytic structure of scattering amplitudes in N=4 SYM.

Abstract

We introduce the notion of Multichannel Conformal Blocks relevant for the Operator Product Expansion for Null Polygon Wilson loops with more than six edges. As an application of these, we decompose the one loop heptagon Wilson loop and predict the value of its two loop OPE discontinuities. At the functional level, the OPE discontinuities are roughly half of the full result. Using symbols they suffice to predict the full two loop result. We also present several new predictions for the heptagon result at any loop order.

Figures (6)

• Figure 1: The octagon NPWL in $\mathbb{R}^{1,1}$ kinematics. (a) The Euclidean picture where all points are spacelike or null separated. This is the kinematical regime considered in this paper. (b) A more physically intuitive Lorentzian picture. The results for the two pictures are related by analytic continuation.
• Figure 2: Octagon ratio $r$. At one loop this ratio is computed by the disconnected correlator of two rectangle Wilson loops. In the OPE picture excitations are produced in the bottom rectangle and absorbed by the top rectangle; in between they propagate freely since at this loop order there are no interactions with the flux tube. That is, the gluon "doesn't know" about the reference square.
• Figure 3: A five point function of local operators. It is the correlation function analog of the heptagon NPWL considered in this paper. For the five point function, the generalized conformal blocks describe the propagation of two primaries and their conformal descendants in two channels simultaneously. In the figure, the two primaries are parametrize by their dimension ($\Delta$) and spin ($l$).
• Figure 4: (a) Finite, conformal invariant ratio of Wilson loops using a single reference square. (b) Finite, conformal invariant ratio of Wilson loops using two reference squares.
• Figure 5: At one loop $\widetilde{r}$ is given by the correlation function of two Wilson loops as represented in the figure. The $u_i$ are given by $u_{1,\dots,7}= \left\{\frac{x_{2,7}^2 x_{3,6}^2}{x_{2,6}^2 x_{3,7}^2},\frac{x_{1,3}^2 x_{4,7}^2}{x_{1,4}^2 x_{3,7}^2},\frac{x_{1,5}^2 x_{2,4}^2}{x_{1,4}^2 x_{2,5}^2},\frac{x_{2,6}^2 x_{3,5}^2}{x_{2,5}^2 x_{3,6}^2},\frac{x_{3,7}^2 x_{4,6}^2}{x_{3,6}^2 x_{4,7}^2},\frac{x_{1,4}^2 x_{5,7}^2}{x_{1,5}^2 x_{4,7}^2},\frac{x_{1,6}^2 x_{2,5}^2}{x_{1,5}^2 x_{2,6}^2}\right\}$; only six of them are independent.