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2017

All-loop Mondrian Diagrammatics and 4-particle Amplituhedron

An, Yang, Li, Yi, Li, Zhinan et al.

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Based on 1712.09990 which handles the 4-particle amplituhedron at 3-loop, we have found an extremely simple pattern, yet far more non-trivial than one might naturally expect: the all-loop Mondrian diagrammatics.

By further simplifying and rephrasing the key relation of positivity in the amplituhedron setting, remarkably, we find a completeness relation unifying all diagrams of the Mondrian types for the 4-particle integrand of planar N=4 SYM to all loop orders, each of which can be mapped to a simple product following a few plain rules designed for this relation.
The explicit examples we investigate span from 3-loop to 7-loop order, and based on them, we classify the basic patterns of Mondrian diagrams into four types: the ladder, cross, brick-wall and spiral patterns.
Interestingly, for some special combinations of ordered subspaces (a concept defined in the previous work), we find failed exceptions of the completeness relation which are called "anomalies", nevertheless, they substantially give hints on the all-loop recursive proof of this relation.

All-loop Mondrian Diagrammatics and 4-particle Amplituhedron · Around

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Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, “The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory,” Phys. Rev. D 75
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Z. Bern, J. J. M. Carrasco, H. Johansson and D. A. Kosower, “Maximally supersymmetric planar Yang-Mills amplitudes at five loops,” Phys. Rev. D 76
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J. Rao, “4-particle Amplituhedron at 3-loop and its Mondrian Diagrammatic Implication,” arXiv:1712.09990 [hep-th]
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https://en.wikipedia.org/wiki/Dihedral_group
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https://en.wikipedia.org/wiki/Separable_permutation , https://en.wikipedia.org/wiki/Schr%C3%B6der_number
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Then

N. Arkani-Hamed and J. Trnka, “Into the Amplituhedron,” JHEP 1412
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