Multi-loop positivity of the planar $$ \mathcal{N} $$ = 4 SYM six-point amplitude
Abstract
We study the six-point NMHV ratio function in planar $$ \mathcal{N} $$ = 4 SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a “radial” direction. We also investigate positivity of the MHV six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.
- Authors:
-
- SLAC National Accelerator Lab., Menlo Park, CA (United States)
- Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
- Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics (QMAP)
- Publication Date:
- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP)
- OSTI Identifier:
- 1334238
- Report Number(s):
- SLAC-PUB-16873
Journal ID: ISSN 1029-8479; arXiv:1611.08325
- Grant/Contract Number:
- AC02-76SF00515
- Resource Type:
- Journal Article: Accepted Manuscript
- Journal Name:
- Journal of High Energy Physics (Online)
- Additional Journal Information:
- Journal Volume: 2017; Journal Issue: 2; Journal ID: ISSN 1029-8479
- Publisher:
- Springer Berlin
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; HEPTH, MATH
Citation Formats
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., and Trnka, Jaroslav. Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude. United States: N. p., 2017.
Web. doi:10.1007/JHEP02(2017)112.
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., & Trnka, Jaroslav. Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude. United States. https://doi.org/10.1007/JHEP02(2017)112
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., and Trnka, Jaroslav. 2017.
"Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude". United States. https://doi.org/10.1007/JHEP02(2017)112. https://www.osti.gov/servlets/purl/1334238.
@article{osti_1334238,
title = {Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude},
author = {Dixon, Lance J. and von Hippel, Matt and McLeod, Andrew J. and Trnka, Jaroslav},
abstractNote = {We study the six-point NMHV ratio function in planar $ \mathcal{N} $ = 4 SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a “radial” direction. We also investigate positivity of the MHV six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.},
doi = {10.1007/JHEP02(2017)112},
url = {https://www.osti.gov/biblio/1334238},
journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 2,
volume = 2017,
place = {United States},
year = {Wed Feb 22 00:00:00 EST 2017},
month = {Wed Feb 22 00:00:00 EST 2017}
}
Web of Science
Works referenced in this record:
Supersymmetric Yang-Mills theories
journal, March 1977
- Brink, Lars; Schwarz, John H.; Scherk, J.
- Nuclear Physics B, Vol. 121, Issue 1
Supersymmetry, supergravity theories and the dual spinor model
journal, April 1977
- Gliozzi, F.; Scherk, J.; Olive, D.
- Nuclear Physics B, Vol. 122, Issue 2
The Amplituhedron
journal, October 2014
- Arkani-Hamed, Nima; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2014, Issue 10
Into the amplituhedron
journal, December 2014
- Arkani-Hamed, Nima; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2014, Issue 12
The amplituhedron from momentum twistor diagrams
journal, February 2015
- Bai, Yuntao; He, Song
- Journal of High Energy Physics, Vol. 2015, Issue 2
Anatomy of the amplituhedron
journal, March 2015
- Franco, Sebastián; Galloni, Daniele; Mariotti, Alberto
- Journal of High Energy Physics, Vol. 2015, Issue 3
The amplituhedron and the one-loop Grassmannian measure
journal, January 2016
- Bai, Yuntao; He, Song; Lam, Thomas
- Journal of High Energy Physics, Vol. 2016, Issue 1
Towards the amplituhedron volume
journal, March 2016
- Ferro, Livia; Lukowski, Tomasz; Orta, Andrea
- Journal of High Energy Physics, Vol. 2016, Issue 3
Iterated path integrals
journal, January 1977
- Chen, Kuo-Tsai
- Bulletin of the American Mathematical Society, Vol. 83, Issue 5
Analytic treatment of the two loop equal mass sunrise graph
journal, January 2005
- Laporta, S.; Remiddi, E.
- Nuclear Physics B, Vol. 704, Issue 1-2
A second-order differential equation for the two-loop sunrise graph with arbitrary masses
journal, January 2012
- Müller-Stach, Stefan; Weinzierl, Stefan; Zayadeh, Raphael
- Communications in Number Theory and Physics, Vol. 6, Issue 1
Uniqueness of two-loop master contours
journal, October 2012
- Caron-Huot, Simon; Larsen, Kasper J.
- Journal of High Energy Physics, Vol. 2012, Issue 10
From the holomorphic Wilson loop to ‘d log’ loop-integrands of super-Yang-Mills amplitudes
journal, May 2013
- Lipstein, Arthur E.; Mason, Lionel
- Journal of High Energy Physics, Vol. 2013, Issue 5
From d logs to dilogs; the super Yang-Mills MHV amplitude revisited
journal, January 2014
- Lipstein, Arthur E.; Mason, Lionel
- Journal of High Energy Physics, Vol. 2014, Issue 1
Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010
- Goncharov, A. B.; Spradlin, M.; Vergu, C.
- Physical Review Letters, Vol. 105, Issue 15
Motivic amplitudes and cluster coordinates
journal, January 2014
- Golden, J. K.; Goncharov, A. B.; Spradlin, M.
- Journal of High Energy Physics, Vol. 2014, Issue 1
A cluster bootstrap for two-loop MHV amplitudes
journal, February 2015
- Golden, John; Spradlin, Marcus
- Journal of High Energy Physics, Vol. 2015, Issue 2
Bootstrapping the three-loop hexagon
journal, November 2011
- Dixon, Lance J.; Drummond, James M.; Henn, Johannes M.
- Journal of High Energy Physics, Vol. 2011, Issue 11
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
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
The four-loop remainder function and multi-Regge behavior at NNLLA in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory
journal, June 2014
- Dixon, Lance J.; Drummond, James M.; Duhr, Claude
- Journal of High Energy Physics, Vol. 2014, Issue 6
The four-loop six-gluon NMHV ratio function
journal, January 2016
- Dixon, Lance J.; von Hippel, Matt; McLeod, Andrew J.
- Journal of High Energy Physics, Vol. 2016, Issue 1
Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
journal, December 2016
- Caron-Huot, Simon; Dixon, Lance J.; McLeod, Andrew
- Physical Review Letters, Vol. 117, Issue 24
A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon
journal, March 2015
- Drummond, J. M.; Papathanasiou, G.; Spradlin, M.
- Journal of High Energy Physics, Vol. 2015, Issue 3
Positive amplitudes in the amplituhedron
journal, August 2015
- Arkani-Hamed, Nima; Hodges, Andrew; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2015, Issue 8
Transcendentality and crossing
journal, January 2007
- Beisert, Niklas; Eden, Burkhard; Staudacher, Matthias
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2007, Issue 01
An exact formula for the radiation of a moving quark in $\mathcal{N} = 4$ super Yang Mills
journal, June 2012
- Correa, Diego; Henn, Johannes; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2012, Issue 6
The four-loop cusp anomalous dimension in $ \mathcal{N} $ = 4 super Yang-Mills and analytic integration techniques for Wilson line integrals
journal, September 2013
- Henn, Johannes M.; Huber, Tobias
- Journal of High Energy Physics, Vol. 2013, Issue 9
Hexagon functions and the three-loop remainder function
journal, December 2013
- Dixon, Lance J.; Drummond, James M.; von Hippel, Matt
- Journal of High Energy Physics, Vol. 2013, Issue 12
Hexagon Wilson loop = six-gluon MHV amplitude
journal, July 2009
- Drummond, J. M.; Henn, J.; Korchemsky, G. P.
- Nuclear Physics B, Vol. 815, Issue 1-2
Optical theorems and steinmann relations
journal, April 1975
- Cahill, Kevin E.; Stapp, Henry P.
- Annals of Physics, Vol. 90, Issue 2
Thermodynamic bubble ansatz
journal, September 2011
- Alday, Luis F.; Gaiotto, Davide; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2011, Issue 9
Spacetime and Flux Tube -Matrices at Finite Coupling for Supersymmetric Yang-Mills Theory
journal, August 2013
- Basso, Benjamin; Sever, Amit; Vieira, Pedro
- Physical Review Letters, Vol. 111, Issue 9
Dual superconformal symmetry of scattering amplitudes in super-Yang–Mills theory
journal, March 2010
- Drummond, J. M.; Henn, J.; Korchemsky, G. P.
- Nuclear Physics B, Vol. 828, Issue 1-2
A current algebra for some gauge theory amplitudes
journal, November 1988
- Nair, V. P.
- Physics Letters B, Vol. 214, Issue 2
Non-MHV Tree Amplitudes in Gauge Theory
journal, July 2004
- Georgiou, George; Glover, E. W. N.; Khoze, Valentin V.
- Journal of High Energy Physics, Vol. 2004, Issue 07
What is the simplest quantum field theory?
journal, September 2010
- Arkani-Hamed, Nima; Cachazo, Freddy; Kaplan, Jared
- Journal of High Energy Physics, Vol. 2010, Issue 9
Pulling the straps of polygons
journal, December 2011
- Gaiotto, Davide; Maldacena, Juan; Sever, Amit
- Journal of High Energy Physics, Vol. 2011, Issue 12
Space-time S-matrix and flux-tube S-matrix IV. Gluons and fusion
journal, September 2014
- Basso, Benjamin; Sever, Amit; Vieira, Pedro
- Journal of High Energy Physics, Vol. 2014, Issue 9
Hexagon OPE resummation and multi-Regge kinematics
journal, February 2016
- Drummond, J. M.; Papathanasiou, G.
- Journal of High Energy Physics, Vol. 2016, Issue 2
Two-loop master integrals for jets: the planar topologies
journal, May 2001
- Gehrmann, T.; Remiddi, E.
- Nuclear Physics B, Vol. 601, Issue 1-2
Harmonic Polylogarithms
journal, February 2000
- Remiddi, E.; Vermaseren, J. A. M.
- International Journal of Modern Physics A, Vol. 15, Issue 05
HPL, a Mathematica implementation of the harmonic polylogarithms
journal, February 2006
- Maître, D.
- Computer Physics Communications, Vol. 174, Issue 3
Systematics of the cusp anomalous dimension
journal, November 2012
- Henn, Johannes M.; Huber, Tobias
- Journal of High Energy Physics, Vol. 2012, Issue 11
The Multiple Zeta Value data mine
journal, March 2010
- Blümlein, J.; Broadhurst, D. J.; Vermaseren, J. A. M.
- Computer Physics Communications, Vol. 181, Issue 3
Numerical evaluation of multiple polylogarithms
journal, May 2005
- Vollinga, Jens; Weinzierl, Stefan
- Computer Physics Communications, Vol. 167, Issue 3
An operator product expansion for polygonal null Wilson loops
journal, April 2011
- Alday, Luis F.; Gaiotto, Davide; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2011, Issue 4
Works referencing / citing this record:
Yangian invariants and cluster adjacency in $$ \mathcal{N} $$ = 4 Yang-Mills
journal, October 2019
- Mago, Jorge; Schreiber, Anders; Spradlin, Marcus
- Journal of High Energy Physics, Vol. 2019, Issue 10
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
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
Rationalizing Loop Integration
text, January 2018
- Bourjaily, Jacob L.; McLeod, Andrew J.; von Hippel, Matt
- arXiv
The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
text, January 2019
- Golden, John; McLeod, Andrew J.; Spradlin, Marcus
- arXiv
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