The Loop Momentum Amplituhedron

TL;DR

This paper introduces the loop momentum amplituhedron, a geometric object encoding tree and loop scattering amplitudes in supersymmetric Yang-Mills theory, providing a new global perspective on loop momenta.

Contribution

It defines the loop momentum amplituhedron as a new geometric structure that captures both tree and loop amplitudes in a unified framework, extending previous positive space concepts.

Findings

Defined the extended positive space for loop amplitudes

Constructed a map to the loop momentum amplituhedron

Provided explicit forms of the canonical differential form

Abstract

In this paper we focus on scattering amplitudes in maximally supersymmetric Yang-Mills theory and define a long sought-after geometry, the loop momentum amplituhedron, which we conjecture to encode tree and (the integrands of) loop amplitudes in spinor helicity variables. Motivated by the structure of amplitude singularities, we define an extended positive space, which enhances the Grassmannian space featuring at tree level, and a map which associates to each of its points tree-level kinematic variables and loop momenta. The image of this map is the loop momentum amplituhedron. Importantly, our formulation provides a global definition of the loop momenta. We conjecture that for all multiplicities and helicity sectors, there exists a canonical logarithmic differential form defined on this space, and provide its explicit form in a few examples.