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.
Highlights
Another remarkable property of scattering amplitudes in planar N = 4 SYM is that they are dual to null polygonal Wilson loops in a dual spacetime [4,5,6,7,8,9, 12]
We propose a new diagrammatic formulation of the all-loop scattering amplitudes/Wilson loops in planar N = 4 SYM, dubbed the “momentum-twistor diagrams”
We propose “momentum-twistor diagrams” as a new diagrammatic representation of all-loop amplitudes/Wilson loops in planar N = 4 SYM
Summary
We start by presenting fundamental ingredients for on-shell diagrams in momentum-twistor space. A generic on-shell diagram consists of trivalent white and black vertices connected by external and internal edges, all drawn on a disk. Each internal edge of the diagram is associated with a momentum twistor Z which is integrated over with the measure d3|4Z = d4|4Z . Each white vertex with three (internal or external) twistors Za, Zb, Zc is represented by an integral over C ∈ G(1, 3) (a 1 × 3 matrix with GL(1) redundancies). Each black vertex with three (internal or external) twistors Za, Zb, Zc is represented by an integral over C ∈ G(2, 3) (a 2 × 3 matrix up to GL(2) redundancies, with minors defined as (i j) ≡ C1,iC2,j − C2,iC1,j), B(a, b, c) =. There are degenerate cases: a black vertex with two edges can be deleted from the diagram, with the two edges identified to be one edge (the two twistors are identified), and a black vertex connected to the boundary by one external edge can be deleted, making the diagram independent of the corresponding external twistor
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