Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs

Nima Arkani-Hamed,Jaroslav Trnka,Cameron Langer,Akshay Yelleshpur Srikant

doi:10.1103/physrevlett.122.051601

Nima Arkani-Hamed, Jaroslav Trnka + Show 2 more

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https://doi.org/10.1103/physrevlett.122.051601

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Abstract

In this Letter we compute a canonical set of cuts of the integrand for maximally helicity violating amplitudes in planar N=4 supersymmetric Yang-Mills theory, where all internal propagators are put on shell. These "deepest cuts" probe the most complicated Feynman diagrams and on-shell processes that can possibly contribute to the amplitude, but are also naturally associated with remarkably simple geometric facets of the amplituhedron. The recent reformulation of the amplituhedron in terms of combinatorial geometry directly in the kinematic (momentum-twistor) space plays a crucial role in understanding this geometry and determining the cut. This provides us with the first nontrivial results on scattering amplitudes in the theory valid for arbitrarily many loops and external particle multiplicities.

Highlights

In this Letter we compute a canonical set of cuts of the integrand for maximally helicity violating amplitudes in planar N 1⁄4 4 supersymmetric Yang-Mills theory, where all internal propagators are put on shell
The original definition was based on a generalization of the positive Grassmannian, centrally connected to on-shell diagrams [2] and loop level recursion relations [3]
The amplituhedron has been extensively studied from many angles including mathematical aspects [5,6,7], positive geometry and volume interpretation [8,9], triangulations [10,11,12,13], connections to on-shell diagrams [14,15], and geometric structures in the final amplitudes [16,17]

Summary

Introduction

In this Letter we compute a canonical set of cuts of the integrand for maximally helicity violating amplitudes in planar N 1⁄4 4 supersymmetric Yang-Mills theory, where all internal propagators are put on shell. All tree-level amplitudes and loop integrands in this theory correspond to the differential forms with logarithmic singularities on boundaries of the amplituhedron geometry.

Results

Conclusion