4-particle Amplituhedron at 3-loop and its Mondrian Diagrammatic Implication

TL;DR

This paper calculates the 4-particle amplituhedron at 3-loop order, revealing a connection between positivity conditions and Mondrian diagrams, with implications for all-loop integrands in planar N=4 SYM.

Contribution

It introduces practical calculation tricks for the 4-particle amplituhedron at 3-loop and uncovers a pattern linking positivity and Mondrian diagrams, suggesting a generalization for all loops.

Findings

Direct calculation of 4-particle amplituhedron at 3-loop

Identification of a pattern linking positivity and Mondrian diagrams

Potential for generalization to all loop orders in N=4 SYM

Abstract

This article provides a direct calculation of the 4-particle amplituhedron at 3-loop order, by introducing a set of practical tricks. After delicately rearranging each piece of this calculation, we find a suggestive connection between positivity conditions and Mondrian diagrams, which will be quantitatively defined. Such a pattern can be generalized for all Mondrian diagrams among all those contribute to the 4-particle integrand of planar N=4 SYM to all loop orders, as a subsequent work 1712.09994 will show.