skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Six-Gluon amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops

Abstract

We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.

Authors:
 [1];  [2];  [3];  [4];  [5];  [6]
  1. McGill Univ., Montreal, QC (Canada). Dept of Physics
  2. SLAC National Accelerator Lab., Menlo Park, CA (United States); Univ. of California, Santa Barbara, CA (United States). Kavli Inst. for Theoretical Physics; Humboldt Univ. of Berlin (Germany). Inst. for Physics
  3. SLAC National Accelerator Lab., Menlo Park, CA (United States)
  4. Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada); Univ. of Copenhagen (Denmark). The Niels Bohr Inst.
  5. SLAC National Accelerator Lab., Menlo Park, CA (United States); Univ. of California, Santa Barbara, CA (United States). Kavli Inst. for Theoretical Physics; Univ. of Copenhagen (Denmark). The Niels Bohr Inst.
  6. Univ. of California, Santa Barbara, CA (United States). Kavli Inst. for Theoretical Physics; Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1562503
Report Number(s):
SLAC–PUB–17413
Journal ID: ISSN 1029-8479
Grant/Contract Number:  
AC02-76SF00515
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: 2019; 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; Scattering Amplitudes; Supersymmetric Gauge Theory

Citation Formats

Caron-Huot, Simon, Dixon, Lance J., Dulat, Falko, von Hippel, Matt, McLeod, Andrew J., and Papathanasiou, Georgios. Six-Gluon amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops. United States: N. p., 2019. Web. doi:10.1007/jhep08(2019)016.
Caron-Huot, Simon, Dixon, Lance J., Dulat, Falko, von Hippel, Matt, McLeod, Andrew J., & Papathanasiou, Georgios. Six-Gluon amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops. United States. doi:10.1007/jhep08(2019)016.
Caron-Huot, Simon, Dixon, Lance J., Dulat, Falko, von Hippel, Matt, McLeod, Andrew J., and Papathanasiou, Georgios. Mon . "Six-Gluon amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops". United States. doi:10.1007/jhep08(2019)016. https://www.osti.gov/servlets/purl/1562503.
@article{osti_1562503,
title = {Six-Gluon amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops},
author = {Caron-Huot, Simon and Dixon, Lance J. and Dulat, Falko and von Hippel, Matt and McLeod, Andrew J. and Papathanasiou, Georgios},
abstractNote = {We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.},
doi = {10.1007/jhep08(2019)016},
journal = {Journal of High Energy Physics (Online)},
number = 8,
volume = 2019,
place = {United States},
year = {2019},
month = {8}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Save / Share:

Works referenced in this record:

The Multi-Regge limit of NMHV amplitudes in N=4 SYM theory
journal, January 2013

  • Lipatov, Lev; Prygarin, Alexander; Schnitzer, Howard J.
  • Journal of High Energy Physics, Vol. 2013, Issue 1
  • DOI: 10.1007/JHEP01(2013)068

Polylogarithmes multiples uniformes en une variable
journal, April 2004


Mellin amplitudes for dual conformal integrals
journal, August 2012

  • Paulos, Miguel F.; Spradlin, Marcus; Volovich, Anastasia
  • Journal of High Energy Physics, Vol. 2012, Issue 8
  • DOI: 10.1007/JHEP08(2012)072

A cluster bootstrap for two-loop MHV amplitudes
journal, February 2015


Collinear Limit of Scattering Amplitudes at Strong Coupling
journal, December 2014


Supersymmetry, supergravity theories and the dual spinor model
journal, April 1977


Conformal properties of four-gluon planar amplitudes and Wilson loops
journal, May 2008


The Galois coaction on $\phi^4$ periods
journal, January 2017


N=4 supersymmetric Yang–Mills scattering amplitudes at high energies: the Regge cut contribution
journal, December 2009


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

Bounded Collection of Feynman Integral Calabi-Yau Geometries
journal, January 2019


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

Feynman amplitudes, coaction principle, and cosmic Galois group
journal, January 2017


Hexagon remainder function in the limit of self-crossing up to three loops
journal, April 2012


Wilson loop remainder function for null polygons in the limit of self-crossing
journal, May 2011


Adjoint BFKL at finite coupling: a short-cut from the collinear limit
journal, January 2015

  • Basso, Benjamin; Caron-Huot, Simon; Sever, Amit
  • Journal of High Energy Physics, Vol. 2015, Issue 1
  • DOI: 10.1007/JHEP01(2015)027

Galois symmetries of fundamental groupoids and noncommutative geometry
journal, June 2005


Fermionic pentagons and NMHV hexagon
journal, May 2015


Local spacetime physics from the Grassmannian
journal, January 2011

  • Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.
  • Journal of High Energy Physics, Vol. 2011, Issue 1
  • DOI: 10.1007/JHEP01(2011)108

Magic identities for conformal four-point integrals
journal, January 2007

  • Drummond, James M.; Henn, Johannes; Smirnov, Vladimir A.
  • Journal of High Energy Physics, Vol. 2007, Issue 01
  • DOI: 10.1088/1126-6708/2007/01/064

Gluon scattering amplitudes at strong coupling
journal, June 2007


Non-MHV Tree Amplitudes in Gauge Theory
journal, July 2004


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

Cluster adjacency beyond MHV
journal, March 2019

  • Drummond, James; Foster, Jack; Gürdoğan, Ömer
  • Journal of High Energy Physics, Vol. 2019, Issue 3
  • DOI: 10.1007/JHEP03(2019)086

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
  • DOI: 10.1007/JHEP03(2015)072

Cluster adjacency and the four-loop NMHV heptagon
journal, March 2019

  • Drummond, James; Foster, Jack; Gürdoğn, Ömer
  • Journal of High Energy Physics, Vol. 2019, Issue 3
  • DOI: 10.1007/JHEP03(2019)087

Hexagon OPE resummation and multi-Regge kinematics
journal, February 2016

  • Drummond, J. M.; Papathanasiou, G.
  • Journal of High Energy Physics, Vol. 2016, Issue 2
  • DOI: 10.1007/JHEP02(2016)185

BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> SUSY
journal, January 2012


Evaluating the six-point remainder function near the collinear limit
journal, October 2014


Nonsinglet pentagons and NMHV amplitudes
journal, July 2015


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


Mixed Tate motives over Z
journal, January 2012


Dual superconformal symmetry of scattering amplitudes in super-Yang–Mills theory
journal, March 2010


Generating tree amplitudes in 𝒩 = 4 SYM and 𝒩 = 8 SG
journal, September 2008

  • Bianchi, Massimo; Elvang, Henriette; Freedman, Daniel Z.
  • Journal of High Energy Physics, Vol. 2008, Issue 09
  • DOI: 10.1088/1126-6708/2008/09/063

Supersymmetric Yang-Mills theories
journal, March 1977


The Galois coaction on the electron anomalous magnetic moment
journal, January 2018


Harmonic Polylogarithms
journal, February 2000

  • Remiddi, E.; Vermaseren, J. A. M.
  • International Journal of Modern Physics A, Vol. 15, Issue 05
  • DOI: 10.1142/S0217751X00000367

Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$
journal, January 2009

  • Brown, Francis C. S.
  • Annales scientifiques de l'École normale supérieure, Vol. 42, Issue 3
  • DOI: 10.24033/asens.2099

Optical theorems and steinmann relations
journal, April 1975


The differential of all two-loop MHV amplitudes in $ \mathcal{N}=4 $ Yang-Mills theory
journal, September 2013


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

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

OPE for all helicity amplitudes
journal, August 2015

  • Basso, Benjamin; Caetano, João; Córdova, Lucía
  • Journal of High Energy Physics, Vol. 2015, Issue 8
  • DOI: 10.1007/JHEP08(2015)018

Matrix pentagons
journal, October 2017


Thermodynamic bubble ansatz
journal, September 2011

  • Alday, Luis F.; Gaiotto, Davide; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2011, Issue 9
  • DOI: 10.1007/JHEP09(2011)032

Exciting the GKP string at any coupling
journal, April 2012


Harmonic sums and polylogarithms generated by cyclotomic polynomials
journal, October 2011

  • Ablinger, Jakob; Blümlein, Johannes; Schneider, Carsten
  • Journal of Mathematical Physics, Vol. 52, Issue 10
  • DOI: 10.1063/1.3629472

A current algebra for some gauge theory amplitudes
journal, November 1988


Pulling the straps of polygons
journal, December 2011

  • Gaiotto, Davide; Maldacena, Juan; Sever, Amit
  • Journal of High Energy Physics, Vol. 2011, Issue 12
  • DOI: 10.1007/JHEP12(2011)011

Single-valued harmonic polylogarithms and the multi-Regge limit
journal, October 2012

  • Dixon, Lance J.; Duhr, Claude; Pennington, Jeffrey
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)074

Graphical functions and single-valued multiple polylogarithms
journal, January 2014


Hexagon Wilson loop = six-gluon MHV amplitude
journal, July 2009


On factorization of multiparticle pentagons
journal, August 2015


Transcendentality and crossing
journal, January 2007

  • Beisert, Niklas; Eden, Burkhard; Staudacher, Matthias
  • Journal of Statistical Mechanics: Theory and Experiment, Vol. 2007, Issue 01
  • DOI: 10.1088/1742-5468/2007/01/P01021

Jumpstarting the all-loop S-matrix of planar $ \mathcal{N} = {4} $ super Yang-Mills
journal, July 2012


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
  • DOI: 10.1007/JHEP06(2014)116

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function
journal, January 2019


    Works referencing / citing this record:

    Optical theorems and steinmann relations
    journal, April 1975


    A current algebra for some gauge theory amplitudes
    journal, November 1988


    Supersymmetry, supergravity theories and the dual spinor model
    journal, April 1977


    Supersymmetric Yang-Mills theories
    journal, March 1977


    Polylogarithmes multiples uniformes en une variable
    journal, April 2004


    Conformal properties of four-gluon planar amplitudes and Wilson loops
    journal, May 2008


    Hexagon Wilson loop = six-gluon MHV amplitude
    journal, July 2009


    Dual superconformal symmetry of scattering amplitudes in super-Yang–Mills theory
    journal, March 2010


    Exciting the GKP string at any coupling
    journal, April 2012


    Fermionic pentagons and NMHV hexagon
    journal, May 2015


    Nonsinglet pentagons and NMHV amplitudes
    journal, July 2015


    On factorization of multiparticle pentagons
    journal, August 2015


    Matrix pentagons
    journal, October 2017


    BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> SUSY
    journal, January 2012


    Harmonic sums and polylogarithms generated by cyclotomic polynomials
    journal, October 2011

    • Ablinger, Jakob; Blümlein, Johannes; Schneider, Carsten
    • Journal of Mathematical Physics, Vol. 52, Issue 10
    • DOI: 10.1063/1.3629472

    Non-MHV Tree Amplitudes in Gauge Theory
    journal, July 2004


    Magic identities for conformal four-point integrals
    journal, January 2007

    • Drummond, James M.; Henn, Johannes; Smirnov, Vladimir A.
    • Journal of High Energy Physics, Vol. 2007, Issue 01
    • DOI: 10.1088/1126-6708/2007/01/064

    Gluon scattering amplitudes at strong coupling
    journal, June 2007


    Generating tree amplitudes in 𝒩 = 4 SYM and 𝒩 = 8 SG
    journal, September 2008

    • Bianchi, Massimo; Elvang, Henriette; Freedman, Daniel Z.
    • Journal of High Energy Physics, Vol. 2008, Issue 09
    • DOI: 10.1088/1126-6708/2008/09/063

    N=4 supersymmetric Yang–Mills scattering amplitudes at high energies: the Regge cut contribution
    journal, December 2009


    Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$
    journal, January 2009

    • Brown, Francis C. S.
    • Annales scientifiques de l'École normale supérieure, Vol. 42, Issue 3
    • DOI: 10.24033/asens.2099

    Mixed Tate motives over Z
    journal, January 2012