Light-like polygonal Wilson loops in 3d Chern-Simons and ABJM theory

TL;DR

The paper analyzes light-like polygonal Wilson loops in three-dimensional CS and ABJM theories to two-loop order, uncovering a one-loop cancellation and a two-loop cusp-related UV divergence in CS that triggers anomalous conformal Ward identities. It derives general constraints on the finite parts via these Ward identities and demonstrates that ABJM's four-cusp Wilson loop at two loops shares the same functional form as the N=4 SYM four-point amplitude, with additional matter contributions shaping the result. The analysis combines ladder, vertex, and gauge/ghost topologies and employs dimensional reduction to regulate divergences, supported by Mellin-Barnes techniques for intricate integrals. The findings suggest a deeper Wilson loop/scattering amplitude relation in ABJM and highlight structural parallels between 3d ABJM and 4d N=4 SYM in these nonlocal observables.

Abstract

We study light-like polygonal Wilson loops in three-dimensional Chern-Simons and ABJM theory to two-loop order. For both theories we demonstrate that the one-loop contribution to these correlators cancels. For pure Chern-Simons, we find that specific UV divergences arise from diagrams involving two cusps, implying the loss of finiteness and topological invariance at two-loop order. Studying those UV divergences we derive anomalous conformal Ward identities for n-cusped Wilson loops which restrict the finite part of the latter to conformally invariant functions. We also compute the four-cusp Wilson loop in ABJM theory to two-loop order and find that the result is remarkably similar to that of the corresponding Wilson loop in N=4 SYM. Finally, we speculate about the existence of a Wilson loop/scattering amplitude relation in ABJM theory.

Figures (7)

• Figure 1: One-loop diagrams.
• Figure 2: One-loop contributions $I_{i+2,i}$ and $I_{i+3,i}$ to the hexagonal Wilson loop.
• Figure 3: Planar two-loop topologies appearing in the polygonal Wilson loop in CS theory. Diagrams where one propagator is connected to a single edge or to two adjacent edges vanish in our gauge and are not displayed.
• Figure 4: The divergence in the vertex diagram arises from the integration region where $s_1 \rightarrow 1, {s}_3 \rightarrow 0$ (and $\beta_{1} \rightarrow 0, \beta_{3} \rightarrow 0$), see equation (\ref{['delta-vertex']}).
• Figure 5: Lagrangian insertion contributing to the Ward identities at one loop.