The Geometry of BCFW for ABJM Loop Integrands

TL;DR

This work develops a loop-level geometric framework for ABJM amplitudes by modeling loop variables as a fiber over tree-level chambers in dual space, introducing L-loop half-chambers that correspond bijectively to L-loop cubic Feynman diagrams with $n/2$ particles. The authors show that the ABJM loop integrand can be computed by summing over all such half-chambers, mirroring the loop-level BCFW recursion and providing a geometric interpretation of these recursions via triangulations of punctured polygons. A striking result is the direct correspondence between ABJM loop half-chambers and $ ext{Tr}(eta^3)$ Feynman diagrams, offering a bridge between ABJM amplitudes and a cubic scalar theory, further reflected in a loop-level Berends-Giele recursion correspondence. The paper also presents explicit examples at six- and eight-points and discusses the prospects and limitations of loop-chamber classifications, including extensions to $ ext{N}=4$ SYM, negative geometries, and BDS-like integrands for ABJM.

Abstract

In this paper we investigate the loop-level geometry of ABJM theory from the perspective of lightcone geometries in dual space. This geometry admits a natural fibration, where one of the loop variables can be naturally interpreted as living in a fiber for each fixed point of a lower-loop geometry. When varying the latter, this leads us to the definition of $L$-loop half-chambers, defined such that `half' of the $(L+1)$-loop fiber remains unchanged. We provide a full classification of these half-chambers, and demonstrate a surprising bijection between $n$-point $L$-loop half-chambers and $L$-loop Feynman diagrams for a cubic scalar theory with $n/2$ particles. Consequently, the sum over $L$-loop half-chambers that computes the $n$-point ABJM amplitude is in direct correspondence with the sum over $L$-loop Feynman diagrams that computes the $(n/2)$-point amplitude of $\text{Tr}(φ^3)$ theory. These Feynman diagrams are also realised geometrically in the structure of the loop fibers. Furthermore, we argue that the half-chamber expansion is equivalent to the loop-level BCFW recursion for ABJM, which arises naturally from our geometric construction. Finally, we will illustrate how $L$-loop chambers emerge as the intersection of two $L$-loop half-chambers, and we provide concrete examples of this construction.

Figures (15)

• Figure 1: A representative of the one-loop fibers for four and six-points, respectively.
• Figure 2: The four tree-level chambers for $n=8$ labelled by a pair of odd and even triangulations of the $n$-gon.
• Figure 3: The one-loop fiber $\Delta_{T_{22}}({\bf x})$ for $n=8$. The corresponding odd/even triangulations $O_2=(t_{137},t_{357}),\,E_2=(t_{248},t_{468})$ are depicted on the right. We see that the dual of these triangulations are tree-level four-point Feynman diagrams which capture part of the structure of $\Delta_{T_{22}}({\bf x})$.
• Figure 4: The lightcone of the vertex $q^+_{135}$ divides $\Delta_6(\mathbf{x})$ into four regions. The negative part $\Delta_{135}^-$ is highlighted in red on the left, and part of the positive contribution $\Delta_{\overline{13}5}=\Delta(x_1,x_2,x_3,q^+_{135})\subset \Delta_{135}^+$ is highlighted in yellow on the right.
• Figure 5: A local view of the vertex $q^+_{135} \in \Delta({\bf x})$ at six-points where the lightcone of the point $y_1 \in \Delta({\bf x})$ is depicted in red. Depending on the position of $y_1$ its null-cone will intersect in either of the four configurations depicted above. The first diagram corresponds to $(y_1-q^+_{135})^2<0$ whilst the remaining three terms correspond to $(y_1-q^+_{135})^2>0$.