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arXiv · 1812.01168

$\Delta$-Algebra and Scattering Amplitudes

Freddy Cachazo·Nick Early·Alfredo Guevara·Sebastian Mizera
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Abstract

In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of $G(2,n)$ as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the $\Delta$-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the $\Delta$-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to $G(k,n)$.

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Freddy Cachazo, Nick Early, Alfredo Guevara, Sebastian Mizera. 2018-12-04. $\Delta$-Algebra and Scattering Amplitudes. https://doi.org/10.1007/jhep02(2019)005

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