On-Shell Structures of MHV Amplitudes Beyond the Planar Limit

TL;DR

The paper advances a purely on-shell description of non-planar amplitudes in ${\cal N}=4$ SYM by showing that all leading singularities of MHV amplitudes can be expressed as a positive sum of differently ordered planar Parke-Taylor factors, via an extended positivity in $G(2,n)$. It introduces a complete combinatorial labeling of reduced non-planar MHV on-shell diagrams by sets of triplets attached to black vertices and provides explicit Grassmannian representations for the corresponding on-shell functions, including a gauge-fixing factor. The work geometrically derives the $U(1)$ decoupling and Kleiss-Kuijf relations as consequences of positivity and residue structure, and conjectures a unified non-planar picture where loop integrands decompose into $d\log$-forms weighted by color factors and unordered Parke-Taylor amplitudes, with final results expressible as sums of polylogarithms. Overall, it offers a finite, structured framework for non-planar MHV leading singularities and suggests a path toward a non-planar extension of the planar amplitude program.

Abstract

We initiate an exploration of on-shell functions in $\mathcal{N}=4$ SYM beyond the planar limit by providing compact, combinatorial expressions for all leading singularities of MHV amplitudes and showing that they can always be expressed as a positive sum of differently ordered Parke-Taylor tree amplitudes. This is understood in terms of an extended notion of positivity in $G(2,n)$, the Grassmannian of 2-planes in $n$ dimensions: a single on-shell diagram can be associated with many different "positive" regions, of which the familiar positive region associated with planar diagrams is just one example. The decomposition into Parke-Taylor factors is simply a "triangulation" of these extended positive regions. The $U(1)$ decoupling and Kleiss-Kuijf (KK) relations satisfied by the Parke-Taylor amplitudes also follow naturally from this geometric picture. These results suggest that non-planar MHV amplitudes in $\mathcal{N}=4$ SYM at all loop orders can be expressed as a sum of polylogarithms weighted by color factors and (unordered) Parke-Taylor amplitudes.