# All-loop cuts from the Amplituhedron

## TL;DR

This paper develops a systematic approach to understanding the all-loop amplituhedron in $ ext{N}=4$ sYM by initially trivializing mutual positivity constraints and analyzing the resulting geometric configurations, applicable to all multiplicities and loops.

## Contribution

It introduces a method to incorporate mutual positivity constraints into the amplituhedron geometry systematically, starting from trivial cases and building complexity.

## Key findings

- Identified geometric configurations corresponding to amplituhedron faces.
- Derived results valid for all multiplicities and loop orders for MHV configurations.
- Provided a framework to gradually reintroduce mutual positivity constraints.

## Abstract

The definition of the amplituhedron in terms of sign flips involves both one-loop constraints and the "mutual positivity" constraint. To gain an understanding of the all-loop integrand of $\mathcal{N}=4$ sYM requires understanding the crucial role played by mutual positivity. This paper is an attempt towards developing a procedure to introduce the complexities of mutual positivity in a systematic and controlled manner. As the first step in this procedure, we trivialize these constraints and understand the geometry underlying the remaining constraints to all loops and multiplicities. We present a host of configurations which correspond to various faces of the amplituhedron. The results we derive are valid at all multiplicities and loop orders for the maximally helicity violating (MHV) configurations. These include detailed derivations for the results in arXiv:1810.08208 [hep-th]. We conclude by indicating how one might move beyond trivial mutual positivity by presenting a series of configuration which re-introduce it bit by bit.