The Amplituhedron from Momentum Twistor Diagrams

TL;DR

This paper introduces momentum-twistor diagrams as a new, Yangian-invariant diagrammatic framework for all-loop amplitudes and Wilson loops in planar $\,\mathcal{N}=4$ SYM. By formulating on-shell diagrams directly in momentum-twistor space, the authors derive all-loop BCFW recursion with boundary, factorization, and forward-limit contributions, and show how these diagrams triangulate the amplituhedron through $C$ and $D$ matrices and positive coordinates. They provide explicit constructions and examples at tree level (including NMHV and $N^2MHV$) and at loop level (one- and two-loops), unveiling a streamlined Kermit-like representation for one-loop and detailed two-loop cells. The work highlights practical computational advantages, a natural link to amplituhedron geometry, and opens avenues for exploring positivity, non-planarity, and extensions to other theories using momentum-twistor methods.

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.