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 N = 4 SYM, by projecting its external and loop momenta to three dimensions. Focusing on the simplest four-point case, we provide strong evidence that the canonical form of this "projected amplituhedron" gives allloop integrand of 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 N = 4 SYM results. Such all-loop simplifications become most transparent for projected negative geometries, where projected loop variables live in AdS3 space and mutual negativity conditions lead to time orderings; amazingly we find that only bipartite graphs, where each node is either a source or sink in time ordering, survive the projection. Our results suggest an unexpected relation between four-point amplitudes in these two theories.

On the geometry of the orthogonal momentum amplituhedron

“…Furthermore, a projection of the loop amplituhedron was recently proposed to describe loop integrands in ABJM theory at four points [13]. This is similar to tree level for SUSY reduced ABJM, where the orthogonal momentum amplituhedron can be seen as a projection of the momentum amplituhedron.…”

“…There is also a loop-level connection between N = 4 SYM and ABJM theory; the all-loop integrand for the ABJM four-point amplitude can be obtained from the all-loop amplituhedron for N = 4 SYM by projecting to three-dimensional kinematics [13]. However, there is no known analogue of the loop momentum amplituhedron [14] for ABJM theory.…”