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

TL;DR

This paper proposes a novel all-loop reduction framework for the 4-particle amplituhedron in planar N=4 SYM, using positivity and Mondrian diagrammatics to relate higher and lower loop integrals through decoupling and residue conditions.

Contribution

It introduces a simple decoupling method and Mondrian rule to systematically relate all-loop amplituhedra to lower loop cases, revealing new coefficient relations.

Findings

Decoupling loop variables simplifies amplituhedron analysis.

Residue relations connect higher and lower loop integrals.

Explicitly explains 6-loop coefficients +2 and 0.

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.