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.
Highlights
Conveniently given in terms of momentum-twistor variables, introduced by Hodges [11]
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 this paper we studied systematically the rich mathematical structures associated with scattering amplitudes of planar N = 4 SYM at one-loop level
Summary
We begin by reviewing the geometry of the Grassmannian, which is likely familiar to the reader. The Grassmannian is a simple but crucial building block in the construction of amplitudes, and it is important to understand it thoroughly. We will discuss a generalization of the Grassmannian which we call the one-loop Grassmannian and denote as G(k, n; 1), with the latter index indicating the loop-level. While the Grassmannian is the basic building block of tree amplitudes, the one-loop Grassmannian is the building block of one-loop amplitudes. Extensions to any loop level L exist which we may denote as G(k, n; L), but their geometry is much richer and we will not have occasion to discuss them in this article
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