On-Shell Structures of MHV Amplitudes Beyond the Planar Limit

Understand

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.