Bootstrapping Null Polygon Wilson Loops

TL;DR

The paper develops an OPE bootstrap framework to obtain two-loop polygon Wilson loop expressions in $R^{1,1}$ by starting from one-loop seeds and incorporating flux-tube excitations organized by $SL(2,R)$. By analyzing octagon and decagon configurations and enforcing symmetry constraints, the authors derive explicit two-loop remainder functions that agree with prior results and extend the construction to general even number of edges. They argue that these results generalize to other planar conformal gauge theories at two loops and discuss prospects for higher-loop extensions, including three-loop octagons and connections to integrability. The work highlights how OPE, flux-tube dynamics, and symmetry can control perturbative corrections and guide analytic bootstrap of Wilson loops and dual amplitudes.

Abstract

We derive the two loop expressions for polygonal Wilson loops by starting from the one loop expressions and applying an operator product expansion. We do this for polygonal Wilson loops in R^{1,1} and find a result in agreement with previous computations. We also discuss the spectrum of excitations around flux tube that connects two null Wilson lines.

Figures (10)

• Figure 1: We start from a general Wilson loop. We select two null lines, denoted in red. We then act with symmetries that are preserved by these null lines on one of the sides. The two null lines preserve a certain $SL(2)$ symmetry. Selecting a dilatation operator inside this $SL(2)$ amounts to a choice of a reference square, denoted here by the red dashed lines. This allows us to perform an expansion which has the rough form seen on the right. The first term comes from the exchange of the flux tube vacuum. The second from the exchange of a single excitation on the flux tube, the third from two excitations, etc. The denominators can be viewed as normalization factors, as we usually have in the standard OPE.
• Figure 2: Definition of a ratio function that is finite and conformal invariant. It involves the selection of a reference square whose vertices (across a diagonal) coincide with vertices of the original polygon.
• Figure 3: (a) The Octagon null Wilson loop in blue embedded in the Penrose diagram of $R^{1,1}$. The red dashed line is the reference square we start with. (b) A different choice of reference square that is suitable for an OPE expansion in the same channel as in (a). In that example, the two choices of reference squares only differ by the position of the "bottom" cusp at 0 (a) and $-\epsilon$ (b). The corresponding OPE expansion parameters are related by the infinitesimal transformation $\tau\to\tau+{\epsilon \over 2} \,e^{-2\tau}$. We must be able to re-write the OPE expansion after this transformation in the same form as before with the same anomalous dimensions. That is only possible if $\gamma_k(p)$ is independent of $k$. In (c) and (d) we draw the same picture as it would have looked like if the Octagon under consideration was Lorentzian.
• Figure 4: The Decagon null Wilson loop in blue embedded in the Penrose diagram of $R^{1,1}$. The red dashed line is the reference square.
• Figure 5: Null Wilson loop with $n=2m$ edges in blue embedded in the Penrose diagram of $R^{1,1}$. The red dashed line is the reference square.