Positive amplitudes in the amplituhedron

Arkani-Hamed, Nima; Hodges, Andrew; Trnka, Jaroslav

doi:10.1007/JHEP08(2015)030

Title: Positive amplitudes in the amplituhedron

Journal Article · Fri Aug 07 00:00:00 EDT 2015 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/JHEP08(2015)030· OSTI ID:1441131

Arkani-Hamed, Nima ^[1]; Hodges, Andrew ^[2]; Trnka, Jaroslav ^[3]

Princeton Univ., NJ (United States)
Wadham College, University of Oxford (United Kingdom)
California Inst. of Technology (CalTech), Pasadena, CA (United States)

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.

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Research Organization:: California Institute of Technology (CalTech), Pasadena, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), High Energy Physics (HEP)

Grant/Contract Number:: SC0011632; FG02-91ER40654

OSTI ID:: 1441131

Alternate ID(s):: OSTI ID: 1596711

Journal Information:: Journal of High Energy Physics (Online), Vol. 2015, Issue 8; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 51 works

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References (16)

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Δ-algebra and scattering amplitudes Cachazo, Freddy; Early, Nick; Guevara, Alfredo Journal of High Energy Physics, Vol. 2019, Issue 2 https://doi.org/10.1007/jhep02(2019)005	journal	February 2019
All-loop cuts from the Amplituhedron Langer, Cameron; Srikant, Akshay Yelleshpur Journal of High Energy Physics, Vol. 2019, Issue 4 https://doi.org/10.1007/jhep04(2019)105	journal	April 2019
On-shell diagrams and the geometry of planar N < 4 $$ \mathcal{N}<4 $$ SYM theories Benincasa, Paolo; Gordo, David Journal of High Energy Physics, Vol. 2017, Issue 11 https://doi.org/10.1007/jhep11(2017)192	journal	November 2017
Positivity, Grassmannian geometry and simplex-like structures of scattering amplitudes Rao, Junjie Journal of High Energy Physics, Vol. 2017, Issue 12 https://doi.org/10.1007/jhep12(2017)147	journal	December 2017
Amplituhedron meets Jeffrey–Kirwan residue Ferro, Livia; Łukowski, Tomasz; Parisi, Matteo Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 4 https://doi.org/10.1088/1751-8121/aaf3c3	journal	December 2018
Accordiohedra as positive geometries for generic scalar field theories Aneesh, P. B.; Jagadale, Mrunmay; Kalyanapuram, Nikhil Physical Review D, Vol. 100, Issue 10 https://doi.org/10.1103/physrevd.100.106013	journal	November 2019
The cosmic Galois group and extended Steinmann relations for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes Caron-Huot, Simon; Dixon, Lance J.; Dulat, Falko Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2019-03749	text	January 2019
Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD Gehrmann, T.; Henn, J. M.; Lo Presti, N. A. American Physical Society https://doi.org/10.5167/uzh-129837	text	January 2016
The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes Caron-Huot, Simon; Dixon, Lance J.; Dulat, Falko Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2019-02565	text	January 2019
Evidence for a Nonplanar Amplituhedron Bern, Zvi; Herrmann, Enrico; Litsey, Sean arXiv https://doi.org/10.48550/arxiv.1512.08591	text	January 2015
Gravity On-shell Diagrams Herrmann, Enrico; Trnka, Jaroslav arXiv https://doi.org/10.48550/arxiv.1604.03479	text	January 2016
All-loop cuts from the Amplituhedron Langer, Cameron; Srikant, Akshay Yelleshpur arXiv https://doi.org/10.48550/arxiv.1902.05951	text	January 2019
The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes Caron-Huot, Simon; Dixon, Lance J.; Dulat, Falko arXiv https://doi.org/10.48550/arxiv.1906.07116	text	January 2019

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