All-loop Mondrian Reduction of 4-particle Amplituhedron at Positive Infinity

Abstract

This article introduces a systematic framework to understand (not to derive yet) the all-loop 4-particle amplituhedron in planar N=4 SYM, utilizing both positivity and the Mondrian diagrammatics. Its key idea is the simplest one so far: we can decouple one or more sets of loop variables (x,y,z,w) from the rest by just setting these variables to either zero or infinity so that their relevant positivity conditions are trivialized, then the all-loop consistency requires that we get lower loop amplituhedra as "residues". These decoupling relations connect higher loop DCI integrals with the lower ones, enabling us to identify their coefficients starting from the 3-loop case. And surprisingly, the delicate mechanism of this process is the simple Mondrian rule D=X+Y, which forces those visually non-Mondrian DCI integrals to have the correct coefficients such that the amplituhedron can exactly reduce to the lower loop one. Examples cover all DCI integrals at L=3,4,5,6, especially, the subtle 6-loop coefficients +2 and 0 are neatly explained in this way.

arXiv comment19 pages, 14 figures

References in corpus (12)

• The Amplituhedron
• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Two-Loop Four-Gluon Amplitudes in N=4 Super-Yang-Mills
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Into the Amplituhedron
• Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM
• Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps
• The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops
• Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality
• Cutting Deep Into The Amplituhedron
• Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM
• 4-particle Amplituhedronics for 3-5 Loops

Cited by in corpus (3)

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• Non-planar BCFW Grassmannian Geometries
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