Loops and legs: ABJM amplitudes from $f$-graphs

TL;DR

This work develops a systematic program to reconstruct ABJM planar scattering amplitudes directly from the permutation-invariant squared-amplitude generator $F_N$, built from weight-3 bipartite $f$-graphs. By exploiting Yangian invariants, dual-conformal integrand bases, and soft-cut recursion, the authors demonstrate how four-point, six-point, and eight-point amplitudes can be extracted from $F_N$ across multiple loop orders, revealing that the full amplitude data may be encoded in the squared objects much as in ${\cal N}=4$ SYM. Key results include new four-point integrands up to $L=6$, a parity-even extraction of six-point one- and two-loop integrands, and a consistent eight-point tree-level reconstruction, along with a structured relation between $M_n^{(L)}$ and the log of amplitudes via negative geometries. These findings bolster the view that ABJM amplitudes for arbitrary multiplicity and loop order can be reconstructed from squared amplitudes, suggesting deep geometric underpinnings and promising connections to ABJM amplituhedra, correlators, and cusp physics.

Abstract

We initiate a systematic study on how to extract planar integrands of (supersymmetric) scattering amplitudes with $L$ loops and $n$ legs in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory from the recently proposed (bosonic) generating function for squared amplitudes with $N:=n{+}L$ dual points; the latter enjoys a hidden permutation symmetry $S_N$ and is given by a linear combination of weight-$3$ planar $f$-graphs that can be recast as bipartite graphs, which manifest important properties of ABJM amplitudes. We provide evidence that it contains sufficient information to reconstruct individual amplitudes, despite the absence of squared amplitudes at odd loops. The extraction of the four-point amplitude is already non-trivial and closely parallels the extraction of five-point amplitudes in ${\cal N}=4$ super Yang-Mills (SYM) from weight-$4$ $f$-graphs: we comment on this similarity and provide new results for $n=4$ ABJM loop integrand up to $L=6$. For higher multiplicities, based on Yangian invariants (including BCFW building blocks for tree amplitudes) and an appropriate basis of planar dual conformal invariant(DCI) integrands, we disentangle six-point integrands up to two loops and eight-point tree amplitude from the squared amplitudes. Our results suggest that ABJM amplitudes of arbitrary multiplicity and loop order can be reconstructed from squared amplitudes, closely paralleling the role of $f$-graphs in $\mathcal{N}=4$ SYM.

Figures (2)

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Theorems & Definitions (1)

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