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Title: Positive amplitudes in the amplituhedron

Abstract

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. In conclusion, 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.

Authors:
 [1];  [2];  [3]
  1. Princeton Univ., NJ (United States)
  2. Wadham College, University of Oxford (United Kingdom)
  3. California Inst. of Technology (CalTech), Pasadena, CA (United States)
Publication Date:
Research Org.:
California Inst. of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1441131
Grant/Contract Number:  
SC0011632; FG02-91ER40654
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2015; Journal Issue: 8; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Supersymmetric gauge theory; Scattering Amplitudes

Citation Formats

Arkani-Hamed, Nima, Hodges, Andrew, and Trnka, Jaroslav. Positive amplitudes in the amplituhedron. United States: N. p., 2015. Web. doi:10.1007/JHEP08(2015)030.
Arkani-Hamed, Nima, Hodges, Andrew, & Trnka, Jaroslav. Positive amplitudes in the amplituhedron. United States. doi:10.1007/JHEP08(2015)030.
Arkani-Hamed, Nima, Hodges, Andrew, and Trnka, Jaroslav. Fri . "Positive amplitudes in the amplituhedron". United States. doi:10.1007/JHEP08(2015)030. https://www.osti.gov/servlets/purl/1441131.
@article{osti_1441131,
title = {Positive amplitudes in the amplituhedron},
author = {Arkani-Hamed, Nima and Hodges, Andrew and Trnka, Jaroslav},
abstractNote = {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. In conclusion, 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.},
doi = {10.1007/JHEP08(2015)030},
journal = {Journal of High Energy Physics (Online)},
number = 8,
volume = 2015,
place = {United States},
year = {2015},
month = {8}
}

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Cited by: 26 works
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Works referenced in this record:

Into the amplituhedron
journal, December 2014

  • Arkani-Hamed, Nima; Trnka, Jaroslav
  • Journal of High Energy Physics, Vol. 2014, Issue 12
  • DOI: 10.1007/JHEP12(2014)182

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The Amplituhedron
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Dual-conformal regularization of infrared loop divergences and the chiral box expansion
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The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM
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Local integrals for planar scattering amplitudes
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