On-shell structures of MHV amplitudes beyond the planar limit
Author(s)Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy; Postnikov, Alexander; Trnka, Jaroslav
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We initiate an exploration of on-shell functions in 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 [subscript +](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 SYM at all loop orders can be expressed as a sum of polylogarithms weighted by color factors and (unordered) Parke-Taylor amplitudes.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of High Energy Physics
Arkani-Hamed, Nima, Jacob L. Bourjaily, Freddy Cachazo, Alexander Postnikov, and Jaroslav Trnka. “On-Shell Structures of MHV Amplitudes Beyond the Planar Limit.” J. High Energ. Phys. 2015, no. 6 (June 2015).
Final published version