Dualities for Loop Amplitudes of N=6 Chern-Simons Matter Theory

TL;DR

The paper demonstrates that the four-point amplitude in the ABJM N=6 Chern-Simons matter theory vanishes at one loop under dimensional reduction, while the two-loop correction is nonzero and precisely matches the corresponding Wilson loop result, up to a finite constant. By constructing a dual-conformal integral basis and fixing coefficients through generalized unitarity, the authors reveal a close connection between ABJM two-loop amplitudes and the one-loop four-point amplitudes of N=4 super Yang-Mills, extending this analogy through a five-dimensional formalism that makes dual conformal symmetry manifest and simplifies tensor integral calculations. This work provides non-trivial evidence for Wilson loop–amplitude duality in ABJM at two loops and motivates the conjecture that planar ABJM four-point amplitudes at higher loops could be related to N=4 sYM structures via an appropriate mapping of couplings and IR properties. The five-dimensional approach not only facilitates computation but also clarifies the role of dual conformal invariance in constraining loop integrands, suggesting a broader, potentially all-loop framework for ABJM amplitudes. Overall, the results point to deep structural parallels between ABJM and N=4 sYM, governed by dual conformal symmetry and infrared behavior, with practical implications for predicting higher-loop amplitudes.

Abstract

In this paper we study the one- and two-loop corrections to the four-point amplitude of N=6 Chern-Simons matter theory. Using generalized unitarity methods we express the one- and two-loop amplitudes in terms of dual-conformal integrals. Explicit integration by using dimensional reduction gives vanishing one-loop result as expected, while the two-loop result is non-vanishing and matches with the Wilson loop computation. Furthermore, the two-loop correction takes the same form as the one-loop correction to the four-point amplitude of N=4 super Yang-Mills. We discuss possible higher loop extensions of this correspondence between the two theories. As a side result, we extend the method of dimensional reduction for three dimensions to five dimensions where dual conformal symmetry is most manifest, demonstrating significant simplification to the computation of integrals.

Figures (6)

• Figure 1: Integrals that contribute to the two-loop amplitude. Here only the propagators and the positions of the dual space coordinates are shown. The numerator for each integral is chosen such that the it is invariant under conformal symmetry in dual space.
• Figure 2: Dual conformal coordinates for one-loop diagram
• Figure 3: The $s$-channel cut of the one-loop four-point amplitude
• Figure 4: The dual conformal invariant integrands with scalar numerators. For each integrand, the (red) solid lines represent propagators while the (blue) dashed lines stretching between $x_i$ and $x_j$ represent the scalar numerator $x_{ij}^2$.
• Figure 5: (a) Double-$s$-channel cut diagram shows the two-loop diagram can be form by sewing three four-point tree diagrams together. (b) Three-particle cut shows the two-loop diagram can be stuck with two five-point tree diagrams which should vanish individually.