All-loop cuts from the Amplituhedron

Abstract

The definition of the amplituhedron in terms of sign flips involves both one-loop constraints and the "mutual positivity" constraint. To gain an understanding of the all-loop integrand of $\mathcal{N}=4$ sYM requires understanding the crucial role played by mutual positivity. This paper is an attempt towards developing a procedure to introduce the complexities of mutual positivity in a systematic and controlled manner. As the first step in this procedure, we trivialize these constraints and understand the geometry underlying the remaining constraints to all loops and multiplicities. We present a host of configurations which correspond to various faces of the amplituhedron. The results we derive are valid at all multiplicities and loop orders for the maximally helicity violating (MHV) configurations. These include detailed derivations for the results in arXiv:1810.08208 [hep-th]. We conclude by indicating how one might move beyond trivial mutual positivity by presenting a series of configuration which re-introduce it bit by bit.

40 pages, 15 figures, 1 table

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Cited by in corpus (10)

• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• Internal boundaries of the loop amplituhedron
• Emergent unitarity from the amplituhedron