All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron

Abstract

We define a new geometry obtained from the all-loop amplituhedron in $\mathcal{N}=4$ SYM by reducing its four-dimensional external and loop momenta to three dimensions. Focusing on the simplest four-point case, we provide strong evidence that the canonical form of this ``reduced amplituhedron" gives the all-loop integrand of the ABJM four-point amplitude. In addition to various all-loop cuts manifested by the geometry, we present explicitly new results for the integrand up to five loops, which are much simpler than results in $\mathcal{N}=4$ SYM. One of the reasons for such all-loop simplifications is that only a very small fraction of the so-called negative geometries survive the dimensional reduction, which corresponds to bipartite graphs. Our results suggest an unexpected relation between four-point amplitudes in these two theories.

5 pages + appendices, version accepted in PRL

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Cited by in corpus (7)

• Integrated negative geometries in ABJM
• The Loop Momentum Amplituhedron
• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• The two-loop eight-point amplitude in ABJM theory
• Leading singularities and chambers of Correlahedron
• The Aharony-Bergman-Jafferis-Maldacena theory on a circle
• The Geometry of BCFW for ABJM Loop Integrands