Into the Amplituhedron

TL;DR

The paper develops the amplituhedron as a positive-geometry reformulation of planar ${\cal N}=4$ SYM scattering, focusing on the four-pparticle, multi-loop case to show how locality and unitarity emerge from positivity. It constructs loop integrands via positivity-driven triangulations, analyzes a wide range of cuts and faces, and reveals striking structures in the multicollinear and logarithmic regimes. A central theme is the simple, path-dependent behavior of multi-collinear cuts and the motivation to study the log of the amplitude, which exhibits improved IR properties and a clear combinatorial organization. The work lays groundwork for a canonical, geometry-first description of amplitudes, with hints of deeper connections to dual geometric formulations and integrability.

Abstract

We initiate an exploration of the physics and geometry of the amplituhedron, starting with the simplest case of the integrand for four-particle scattering in planar N=4 SYM. We show how the textbook structure of the unitarity double-cut follows from the positive geometry. We also use the geometry to expose the behavior of the multicollinear limit, providing a direct motivation for studying the logarithm of the amplitude. In addition to computing the two and three-loop integrands, we explore various lower-dimensional faces of the amplituhedron, thereby computing non-trivial cuts of the integrand to all loop orders.