Notes on scattering amplitudes as differential forms

• Published in: The journal of high energy physics Vol. 2018; no. 10; pp. 1 - 25
• Main Authors: He, Song, Zhang, Chi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2018; Springer Nature B.V; SpringerOpen
• Subjects: Amplitudes; Canonical forms; Classical and Quantum Gravitation; Differential and Algebraic Geometry; Elementary Particles; Gauge theory; Helicity; High energy physics; Momentum; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Scattering Amplitudes; String Theory; Supersymmetric Gauge Theory; Scattering Amplitudes; Differential and Algebraic Geometry; Supersymmetric Gauge Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP10(2018)054

A bstract Inspired by the idea of viewing amplitudes in N = 4 SYM as differential forms on momentum twistor space, we introduce differential forms on the space of spinor variables, which combine helicity amplitudes in any four-dimensional gauge theory as a single object. In this note we focus on such differential forms in N = 4 SYM, which can also be thought of as “bosonizing” superamplitudes in non-chiral superspace. Remarkably all tree-level amplitudes in N = 4 SYM combine to a d log form in spinor variables, which is given by pushforward of canonical forms of Grassmannian cells. The tree forms can also be obtained using BCFW or inverse-soft construction, and we present all-multiplicity expression for MHV and NMHV forms to illustrate their simplicity. Similarly all-loop planar integrands can be naturally written as d log forms in the Grassmannian/on-shell-diagram picture, and we expect the same to hold beyond the planar limit. Just as the form in momentum twistor space reveals underlying positive geometry of the amplituhedron, the form in terms of spinor variables strongly suggests an “amplituhedron in momentum space”. We initiate the study of its geometry by connecting it to the moduli space of Witten’s twistor-string theory, which provides a pushforward formula for tree forms in N = 4 SYM.