The Amplituhedron from Momentum Twistor Diagrams

Abstract

We propose a new diagrammatic formulation of the all-loop scattering amplitudes/Wilson loops in planar N=4 SYM, dubbed the "momentum-twistor diagrams". These are on-shell-diagrams obtained by gluing trivalent black and white vertices defined in momentum twistor space, which, in the reduced diagram case, are known to be related to diagrams in the original twistor space. The new diagrams are manifestly Yangian invariant, and they naturally represent factorization and forward-limit contributions in the all-loop BCFW recursion relations in momentum twistor space, in a fashion that is completely different from those in momentum space. We show how to construct and evaluate momentum-twistor diagrams, and how to use them to obtain tree-level amplitudes and loop-level integrands; in particular for the latter we identify an isolated bubble-structure for each loop variable, arising from a forward limit, or entangled removal of particles. From a given diagram one can directly read off the C, D matrices via a generalized "boundary measurement"; this in turn determines a cell in the amplituhedron associated with the amplitude, and our diagrammatic representations of the amplitude can provide triangulations of the amplituhedron with generally very intricate geometries. To demonstrate the computational power of the formalism, we give explicit results for general two-loop integrands, and the cells of the complete amplituhedron for two-loop MHV amplitudes.

arXiv comment39 pages, 34 figures

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• The Grassmannian Origin Of Dual Superconformal Invariance
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Cited by in corpus (35)

• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Positive Geometries and Canonical Forms
• Unwinding the Amplituhedron in Binary
• Positive Amplitudes In The Amplituhedron
• Evidence for a Nonplanar Amplituhedron
• Prescriptive Unitarity
• Gravity On-shell Diagrams
• Landau Singularities from the Amplituhedron
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Cutting Deep Into The Amplituhedron
• The Correlahedron
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• The Amplituhedron and the One-loop Grassmannian Measure
• Notes on Scattering Amplitudes as Differential Forms
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Properties of scattering forms and their relation to associahedra
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Grassmannians and form factors with $q^2=0$ in N=4 SYM theory
• Wilson lines, Grassmannians and Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Sign Flip Triangulations of the Amplituhedron
• Four dimensional ambitwistor strings and form factors of local and Wilson line operators
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• The Loop Momentum Amplituhedron
• All-loop cuts from the Amplituhedron
• Non-planar BCFW Grassmannian Geometries
• The all-loop conjecture for integrands of reggeon amplitudes in N=4 SYM
• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• Poles At Infinity in On-shell Diagrams
• The twistor Wilson loop and the amplituhedron
• Learning the Simplicity of Scattering Amplitudes
• Top-forms of Leading Singularities in Nonplanar Multi-loop Amplitudes
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Simplex-like Structures of Maximally Supersymmetric Scattering Amplitudes
• Plabic Tangles and Cluster Promotion Maps