Triangulation-free Trivialization of 2-loop MHV Amplituhedron

TL;DR

This paper presents a universal, triangulation-free method for implementing positivity in the 2-loop n-particle MHV amplituhedron, simplifying the analysis and potentially extending to higher loops, avoiding complex case-by-case triangulations.

Contribution

It introduces a new universal approach to positivity that bypasses traditional triangulation, simplifying the analysis of the 2-loop MHV amplituhedron and exploring potential generalizations.

Findings

The new method is universal for all linear positive conditions.

It revises the proof of the 2-loop n-particle MHV amplituhedron.

Preliminary exploration towards higher loops is presented.

Abstract

This article introduces a new approach to implement positivity for the 2-loop n-particle MHV amplituhedron, circumventing the conventional triangulation with respect to positive variables of each cell carved out by the sign flips. This approach is universal for all linear positive conditions and hence free of case-by-case triangulation, as an application of the trick of positive infinity first introduced in 1910.14612 for the multi-loop 4-particle amplituhedron. Moreover, the proof of 2-loop n-particle MHV amplituhedron in 1812.01822 is revised, and we explain the nontriviality and difficulty of using conventional triangulation while the results have a simple universal pattern. A further example is presented to tentatively explore its generalization towards handling multiple positive conditions at 3-loop and higher.