Positive and Negative Ladders in Loop Space

TL;DR

This paper derives a unified, simple formula for the canonical forms of all-loop ladder contributions in scattering amplitudes within $ ext{N}=4$ SYM, using a novel graphical notation and chamber decomposition.

Contribution

It introduces a new graphical notation for loop space geometries and provides a universal formula for ladder integrals in MHV amplitudes at all loops.

Findings

Unified formula for all-loop ladder integrals.

Graphical notation for loop space chambers.

Ladder contributions as sums over maximal cuts.

Abstract

Motivated by a new term-wise factorised formula for the two-loop MHV integrand for scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills (SYM), together with recent results for the five-point negative ladders in loop space, we present the canonical forms for general ladders in loop space for an arbitrary number of particles to all loops. We make use of the graphical notation introduced in the negative geometries literature, where each loop momentum is represented as a vertex, and mutual positivity (resp. negativity) conditions as a positive (resp. negative) edge. In this paper we extend this notation to include the notion of chambers of the one-loop momentum amplituhedron. Equipped with this new graphical notation, we find the canonical form of the $L$-loop (negative/positive) ladders for all MHV$_n$ amplitudes. Our final formula is remarkably simple and reminiscent of the chiral pentagon expansion of the one and two loop momentum amplituhedron. It expresses ladder contributions as sums over maximal cuts, with each term appearing in the sum factorising into products of either chiral pentagons or their simple generalisations.