Abstract
We present new triangulations of the m = 4 amplituhedron relevant for scattering amplitudes in planar \( \mathcal{N} \) = 4 super-Yang-Mills, obtained directly from the combinatorial definition of the geometry. Using the “sign flip” characterization of the amplituhedron, we reproduce the canonical forms for the all-multiplicity next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (N2MHV) tree-level as well as the NMHV one-loop cases, without using any input from traditional amplitudes methods. Our results provide strong evidence for the equivalence of the original definition of the amplituhedron [1] and the topological one [2], and suggest a new path forward for computing higher loop amplitudes geometrically. In particular, we realize the NMHV one-loop amplituhedron as the intersection of two amplituhedra of lower dimensionality, which is reflected in the novel structure of the corresponding canonical form.
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Kojima, R., Langer, C. Sign flip triangulations of the amplituhedron. J. High Energ. Phys. 2020, 121 (2020). https://doi.org/10.1007/JHEP05(2020)121
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DOI: https://doi.org/10.1007/JHEP05(2020)121