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

TL;DR

This paper introduces a new geometric approach to compute all-loop four-point amplitudes in ABJM theory by reducing the amplituhedron from four to three dimensions, revealing simplified integrands and unexpected relations with ${ m N}=4$ SYM.

Contribution

The paper constructs a reduced amplituhedron geometry in three dimensions that yields the all-loop integrand for ABJM four-point amplitudes, simplifying calculations and uncovering new connections.

Findings

Explicit all-loop integrand results up to five loops for ABJM.

Simplification of integrands due to reduced negative geometries.

Evidence of a relation between ABJM and ${ m N}=4$ SYM amplitudes.

Abstract

We define a new geometry obtained from the all-loop amplituhedron in ${\cal N}=4$ 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 ${\cal N}=4$ 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.