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.
Highlights
Calculating amplitudes or loop integrands starting from the amplituhedron requires the construction of the canonical form associated to the geometry
We present new triangulations of the m = 4 amplituhedron relevant for scattering amplitudes in planar N = 4 super-Yang-Mills, obtained directly from the combinatorial definition of the geometry
The topological characterization of the amplituhedron replaces the computation of scattering amplitudes and loop integrands in planar N = 4 sYM by a simple to state geometry problem
Summary
The original definition of the amplituhedron is a generalization of the interior of plane polygons to the positive Grassmannian [1]. The sign flip definition of the loop-level amplituhedron supplements the tree-level conditions with two kinds of conditions: each loop must be in a copy of the one-loop amplituhedron, and the loops must be mutually positive. This gives the definition for A(n,k, ) to be the space of all (k+2)-planes (Y AB)γ, γ = 1, . The tree-level scattering amplitudes and loop-level integrands of planar N =4 sYM are extracted directly from the canonical forms Ω(n,k, ) defined to have logarithmic singularities on all boundaries of A(n,k, ). Integrates over the four-dimensional Grassmann variables φ1, . . . , φk [1]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
