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On-Shell Structures of MHV Amplitudes Beyond the Planar Limit

Arkani-Hamed, Nima and Bourjaily, Jacob L. and Cachazo, Freddy and Postnikov, Alexander and Trnka, Jaroslav (2015) On-Shell Structures of MHV Amplitudes Beyond the Planar Limit. Journal of High Energy Physics, 2015 (6). Art. No. 179. ISSN 1126-6708. doi:10.1007/JHEP06(2015)179.

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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_+(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.

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Additional Information:Open Access, © 2015 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received: May 16, 2015; Accepted: June 8, 2015; Published: June 25, 2015. We are grateful to Andrew Hodges and Lauren Williams for helpful iscussions. N.A.-H. is supported by the Department of Energy under grant number DE-FG02-91ER40654; J.L.B. is supported by a MOBILEX grant from the Danish Council for Independent Research; F.C. is supported by the Perimeter Institute for Theoretical Physics which is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation; and J.T. is supported in part by the David and Ellen Lee Postdoctoral Scholarship and by the Department of Energy under grant number DE-SC0011632.
Group:Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG02-91ER40654
Danish Council for Independent ResearchMOBILEX
Perimeter Institute for Theoretical PhysicsUNSPECIFIED
Industry CanadaUNSPECIFIED
Province of Ontario, Ministry of Research and InnovationUNSPECIFIED
David and Ellen Lee Postdoctoral ScholarshipUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Subject Keywords:Scattering Amplitudes, 1/N Expansion, Supersymmetric gauge theory
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Issue or Number:6
Record Number:CaltechAUTHORS:20150122-082315937
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:53972
Deposited By: Joy Painter
Deposited On:22 Jan 2015 21:08
Last Modified:10 Nov 2021 20:25

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