The Amplituhedron and the One-loop Grassmannian Measure

Abstract

All-loop planar scattering amplitudes in maximally supersymmetric Yang-Mills theory can be formulated geometrically in terms of the "amplituhedron". We study the mathematical structures of the one-loop amplituhedron, and present a new formula for its canonical measure, or the one-loop Grassmannian measure formula. Using the recently proposed momentum-twistor diagrams, we show that there is a correspondence between the cells of one-loop amplituhedron, BCFW terms or equivalently on-shell diagrams, and residues of the one-loop Grassmannian formula. In particular, for the first non-trivial case of one-loop NMHV, these structures are naturally associated with a nice geometric picture as polygons in projective space, as we discuss in various illustrative examples.

31+10 pages, many figures

References in corpus (5)

• Total positivity, Grassmannians, and networks
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• The Amplituhedron from Momentum Twistor Diagrams
• Grassmannians for scattering amplitudes in 4d $\mathcal{N}=4$ SYM and 3d ABJM

Cited by in corpus (24)

• Positive Geometries and Canonical Forms
• Unwinding the Amplituhedron in Binary
• Evidence for a Nonplanar Amplituhedron
• Prescriptive Unitarity
• Gravity On-shell Diagrams
• Landau Singularities from the Amplituhedron
• Cutting Deep Into The Amplituhedron
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Notes on Scattering Amplitudes as Differential Forms
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• The Loop Momentum Amplituhedron
• All-loop cuts from the Amplituhedron
• Positivity Sectors and the Amplituhedron
• Yangian Symmetry for the Tree Amplituhedron
• On two notions of total positivity for partial flag varieties
• The twistor Wilson loop and the amplituhedron
• Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Bipartite Graphs, On-Shell Diagrams, and Bipartite Field Theories: a Computational Package in Mathematica