# On-shell structures of MHV amplitudes beyond the planar limit

Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander Postnikov, Jaroslav Trnka

Research output: Contribution to journalArticlepeer-review

46 Scopus citations

## Abstract

Abstract: We initiate an exploration of on-shell functions in N=4$$\mathcal{N}=4$$ 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 G+(2, n) 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 N=4$$\mathcal{N}=4$$ SYM at all loop orders can be expressed as a sum of polylogarithms weighted by color factors and (unordered) Parke-Taylor amplitudes.

Original language English (US) 179 Journal of High Energy Physics 2015 6 https://doi.org/10.1007/JHEP06(2015)179 Published - Jun 2 2015

## All Science Journal Classification (ASJC) codes

• Nuclear and High Energy Physics