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Positive Amplitudes In The Amplituhedron

Arkani-Hamed, Nima and Hodges, Andrew and Trnka, Jaroslav (2015) Positive Amplitudes In The Amplituhedron. Journal of High Energy Physics, 2015 (8). Art. No. 30. ISSN 1126-6708. doi:10.1007/JHEP08(2015)030.

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The all-loop integrand for scattering amplitudes in planar N = 4 SYM is determined by an “amplitude form” with logarithmic singularities on the boundary of the amplituhedron. In this note we provide strong evidence for a new striking property of the superamplitude, which we conjecture to be true to all loop orders: the amplitude form is positive when evaluated inside the amplituhedron. The statement is sensibly formulated thanks to the natural “bosonization” of the superamplitude associated with the amplituhedron geometry. However this positivity is not manifest in any of the current approaches to scattering amplitudes, and in particular not in the cellulations of the amplituhedron related to on-shell diagrams and the positive grassmannian. The surprising positivity of the form suggests the existence of a “dual amplituhedron” formulation where this feature would be made obvious. We also suggest that the positivity is associated with an extended picture of amplituhedron geometry, with the amplituhedron sitting inside a co-dimension one surface separating “legal” and “illegal” local singularities of the amplitude. We illustrate this in several simple examples, obtaining new expressions for amplitudes not associated with any triangulations, but following in a more invariant manner from a global view of the positive geometry.

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Additional Information:© 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: February 5, 2015. Accepted: July 13, 2015. Published: August 7, 2015. We thank Jake Bourjaily and Thomas Lam for stimulating discussions. We also thank Simon Caron-Huot for discussions, and for suggesting and checking the positivity of the integrated 6-point ratio function at one loop. N. A.-H. is supported by the Department of Energy under grant number DE-FG02-91ER40654. 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
David and Ellen Lee Postdoctoral ScholarshipUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Subject Keywords:Supersymmetric gauge theory, Scattering Amplitudes
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Issue or Number:8
Record Number:CaltechAUTHORS:20150316-142202703
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Official Citation:Positive amplitudes in the amplituhedron Nima Arkani-Hamed, Andrew Hodges, Jaroslav Trnka Article 30
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:55808
Deposited By: Joy Painter
Deposited On:16 Mar 2015 21:38
Last Modified:10 Nov 2021 20:50

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