Positive amplitudes in the amplituhedron

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

doi:10.1007/JHEP08(2015)030

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:

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)

Publication Date:: Fri Aug 07 00:00:00 EDT 2015

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

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

OSTI Identifier:: 1441131

Alternate Identifier(s):: OSTI ID: 1596711

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.

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                    Arkani-Hamed, Nima, Hodges, Andrew, & Trnka, Jaroslav. Positive amplitudes in the amplituhedron.  United States.  https://doi.org/10.1007/JHEP08(2015)030

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                    Arkani-Hamed, Nima, Hodges, Andrew, and Trnka, Jaroslav. Fri .  
"Positive amplitudes in the amplituhedron".  United States.  https://doi.org/10.1007/JHEP08(2015)030.  https://www.osti.gov/servlets/purl/1441131.

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@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         = {Fri Aug 07 00:00:00 EDT 2015},

  month        = {Fri Aug 07 00:00:00 EDT 2015}

}

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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

Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in SYM
journal, September 2012

Eden, Burkhard; Heslop, Paul; Korchemsky, Gregory P.
Nuclear Physics B, Vol. 862, Issue 2
DOI: 10.1016/j.nuclphysb.2012.04.013

Bootstrapping an NMHV amplitude through three loops
journal, October 2014

Dixon, Lance J.; von Hippel, Matt
Journal of High Energy Physics, Vol. 2014, Issue 10
DOI: 10.1007/JHEP10(2014)065

Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory
journal, January 2012

Dixon, Lance J.; Drummond, James M.; Henn, Johannes M.
Journal of High Energy Physics, Vol. 2012, Issue 1
DOI: 10.1007/JHEP01(2012)024

New recursion relations for tree amplitudes of gluons
journal, May 2005

Britto, Ruth; Cachazo, Freddy; Feng, Bo
Nuclear Physics B, Vol. 715, Issue 1-2
DOI: 10.1016/j.nuclphysb.2005.02.030

The Amplituhedron
journal, October 2014

Arkani-Hamed, Nima; Trnka, Jaroslav
Journal of High Energy Physics, Vol. 2014, Issue 10
DOI: 10.1007/JHEP10(2014)030

Anatomy of the amplituhedron
journal, March 2015

Franco, Sebastián; Galloni, Daniele; Mariotti, Alberto
Journal of High Energy Physics, Vol. 2015, Issue 3
DOI: 10.1007/JHEP03(2015)128

The amplituhedron from momentum twistor diagrams
journal, February 2015

Bai, Yuntao; He, Song
Journal of High Energy Physics, Vol. 2015, Issue 2
DOI: 10.1007/JHEP02(2015)065

Eliminating spurious poles from gauge-theoretic amplitudes
journal, May 2013

Hodges, Andrew
Journal of High Energy Physics, Vol. 2013, Issue 5
DOI: 10.1007/JHEP05(2013)135

A note on polytopes for scattering amplitudes
journal, April 2012

Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.
Journal of High Energy Physics, Vol. 2012, Issue 4
DOI: 10.1007/JHEP04(2012)081

Dual-conformal regularization of infrared loop divergences and the chiral box expansion
journal, January 2015

Bourjaily, Jacob; Caron-Huot, Simon; Trnka, Jaroslav
Journal of High Energy Physics, Vol. 2015, Issue 1
DOI: 10.1007/JHEP01(2015)001

The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM
journal, January 2011

Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.
Journal of High Energy Physics, Vol. 2011, Issue 1
DOI: 10.1007/JHEP01(2011)041

Local integrals for planar scattering amplitudes
journal, June 2012

Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.
Journal of High Energy Physics, Vol. 2012, Issue 6
DOI: 10.1007/JHEP06(2012)125

MHV diagrams in momentum twistor space
journal, December 2010

Bullimore, Mathew; Mason, Lionel; Skinner, David
Journal of High Energy Physics, Vol. 2010, Issue 12
DOI: 10.1007/JHEP12(2010)032

Dual conformal symmetry of 1-loop NMHV amplitudes in $ \mathcal{N} = 4 $ SYM theory
journal, March 2010

Elvang, Henriette; Freedman, Daniel Z.; Kiermaier, Michael
Journal of High Energy Physics, Vol. 2010, Issue 3
DOI: 10.1007/JHEP03(2010)075

The soft-collinear bootstrap: $ \mathcal{N} = {4} $ Yang-Mills amplitudes at six- and seven-loops
journal, March 2012

Bourjaily, J. L.; DiRe, A.; Shaikh, A.
Journal of High Energy Physics, Vol. 2012, Issue 3
DOI: 10.1007/JHEP03(2012)032

Works referencing / citing this record:

Δ-algebra and scattering amplitudes
journal, February 2019

Cachazo, Freddy; Early, Nick; Guevara, Alfredo
Journal of High Energy Physics, Vol. 2019, Issue 2
DOI: 10.1007/jhep02(2019)005

All-loop cuts from the Amplituhedron
journal, April 2019

Langer, Cameron; Srikant, Akshay Yelleshpur
Journal of High Energy Physics, Vol. 2019, Issue 4
DOI: 10.1007/jhep04(2019)105

On-shell diagrams and the geometry of planar N < 4 $$ \mathcal{N}<4 $$ SYM theories
journal, November 2017

Benincasa, Paolo; Gordo, David
Journal of High Energy Physics, Vol. 2017, Issue 11
DOI: 10.1007/jhep11(2017)192

Positivity, Grassmannian geometry and simplex-like structures of scattering amplitudes
journal, December 2017

Rao, Junjie
Journal of High Energy Physics, Vol. 2017, Issue 12
DOI: 10.1007/jhep12(2017)147

Amplituhedron meets Jeffrey–Kirwan residue
journal, December 2018

Ferro, Livia; Łukowski, Tomasz; Parisi, Matteo
Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 4
DOI: 10.1088/1751-8121/aaf3c3

Accordiohedra as positive geometries for generic scalar field theories
journal, November 2019

Aneesh, P. B.; Jagadale, Mrunmay; Kalyanapuram, Nikhil
Physical Review D, Vol. 100, Issue 10
DOI: 10.1103/physrevd.100.106013

The cosmic Galois group and extended Steinmann relations for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes
text, January 2019

Caron-Huot, Simon; Dixon, Lance J.; Dulat, Falko
Deutsches Elektronen-Synchrotron, DESY, Hamburg
DOI: 10.3204/pubdb-2019-03749

Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
text, January 2016

Gehrmann, T.; Henn, J. M.; Lo Presti, N. A.
American Physical Society
DOI: 10.5167/uzh-129837

The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes
text, January 2019

Caron-Huot, Simon; Dixon, Lance J.; Dulat, Falko
Deutsches Elektronen-Synchrotron, DESY, Hamburg
DOI: 10.3204/pubdb-2019-02565

Evidence for a Nonplanar Amplituhedron
text, January 2015

Bern, Zvi; Herrmann, Enrico; Litsey, Sean
arXiv
DOI: 10.48550/arxiv.1512.08591

Gravity On-shell Diagrams
text, January 2016

Herrmann, Enrico; Trnka, Jaroslav
arXiv
DOI: 10.48550/arxiv.1604.03479

All-loop cuts from the Amplituhedron
text, January 2019

Langer, Cameron; Srikant, Akshay Yelleshpur
arXiv
DOI: 10.48550/arxiv.1902.05951

The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes
text, January 2019

Caron-Huot, Simon; Dixon, Lance J.; Dulat, Falko
arXiv
DOI: 10.48550/arxiv.1906.07116

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Similar Records

Title: Positive amplitudes in the amplituhedron

Abstract

Citation Formats

Into the amplituhedron journal, December 2014

Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in SYM journal, September 2012

Bootstrapping an NMHV amplitude through three loops journal, October 2014

Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory journal, January 2012

New recursion relations for tree amplitudes of gluons journal, May 2005

The Amplituhedron journal, October 2014

Anatomy of the amplituhedron journal, March 2015

The amplituhedron from momentum twistor diagrams journal, February 2015

Eliminating spurious poles from gauge-theoretic amplitudes journal, May 2013

A note on polytopes for scattering amplitudes journal, April 2012

Dual-conformal regularization of infrared loop divergences and the chiral box expansion journal, January 2015

The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM journal, January 2011

Local integrals for planar scattering amplitudes journal, June 2012

MHV diagrams in momentum twistor space journal, December 2010

Dual conformal symmetry of 1-loop NMHV amplitudes in $ \mathcal{N} = 4 $ SYM theory journal, March 2010

The soft-collinear bootstrap: $ \mathcal{N} = {4} $ Yang-Mills amplitudes at six- and seven-loops journal, March 2012

Δ-algebra and scattering amplitudes journal, February 2019

All-loop cuts from the Amplituhedron journal, April 2019

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Positivity, Grassmannian geometry and simplex-like structures of scattering amplitudes journal, December 2017

Amplituhedron meets Jeffrey–Kirwan residue journal, December 2018

Accordiohedra as positive geometries for generic scalar field theories journal, November 2019

The cosmic Galois group and extended Steinmann relations for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes text, January 2019

Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD text, January 2016

The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes text, January 2019

Evidence for a Nonplanar Amplituhedron text, January 2015

Gravity On-shell Diagrams text, January 2016

All-loop cuts from the Amplituhedron text, January 2019

The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes text, January 2019

Into the amplituhedron
journal, December 2014

Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in SYM
journal, September 2012

Bootstrapping an NMHV amplitude through three loops
journal, October 2014

Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory
journal, January 2012

New recursion relations for tree amplitudes of gluons
journal, May 2005

The Amplituhedron
journal, October 2014

Anatomy of the amplituhedron
journal, March 2015

The amplituhedron from momentum twistor diagrams
journal, February 2015

Eliminating spurious poles from gauge-theoretic amplitudes
journal, May 2013

A note on polytopes for scattering amplitudes
journal, April 2012

Dual-conformal regularization of infrared loop divergences and the chiral box expansion
journal, January 2015

The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM
journal, January 2011

Local integrals for planar scattering amplitudes
journal, June 2012

MHV diagrams in momentum twistor space
journal, December 2010

Dual conformal symmetry of 1-loop NMHV amplitudes in $ \mathcal{N} = 4 $ SYM theory
journal, March 2010

The soft-collinear bootstrap: $ \mathcal{N} = {4} $ Yang-Mills amplitudes at six- and seven-loops
journal, March 2012

Δ-algebra and scattering amplitudes
journal, February 2019

All-loop cuts from the Amplituhedron
journal, April 2019

On-shell diagrams and the geometry of planar N < 4 $$ \mathcal{N}<4 $$ SYM theories
journal, November 2017

Positivity, Grassmannian geometry and simplex-like structures of scattering amplitudes
journal, December 2017

Amplituhedron meets Jeffrey–Kirwan residue
journal, December 2018

Accordiohedra as positive geometries for generic scalar field theories
journal, November 2019

The cosmic Galois group and extended Steinmann relations for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes
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Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
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The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes
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