Triangulation-free trivialization of 2-loop MHV amplituhedron

Ryota Kojima,Junjie Rao

doi:10.1007/jhep10(2020)140

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 [6] for the multi-loop 4-particle amplituhedron. Moreover, the proof of 2-loop n-particle MHV amplituhedron in [4] 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.

Highlights

Plus a single mutual positive condition, and the relevant background of its integrand can be found in [14,15,16]
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 free of case-by-case triangulation, as an application of the trick of positive infinity first introduced in [6] for the multi-loop 4-particle amplituhedron
We are ready to move forward, to explore the extraordinary simplicity hidden in the 2-loop MHV amplituhedron

Summary

Minimal review of positive d log forms and dimensionless ratios

To get familiar with the mathematical concepts we will extensively use, let’s first give a minimal review of d log forms in positive geometry. As defined in [2], for a positive variable x without further restriction, we know its d log form is dx (2.1). In terms of dimensionless ratios [6], which is a more natural way to characterize positive d log forms. As done in [5], we can generalize these conditions to n xi > a and n xi < a, and the corresponding dimensionless ratios sum to unity:. A can be generalized to a sum of positive variables, the dimensionless ratio of n xi > a = m yj is n xi. Positive infinity is an indispensable notion for fully understanding the cut structure of the loop amplituhedron. We are ready to move forward, to explore the extraordinary simplicity hidden in the 2-loop MHV amplituhedron

Triangulation-free trivialization for linear polynomials

An example of quasi-linear polynomials

Outlook