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

Title: Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

Abstract

We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar N=4 supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory’s two-loop MHV amplitudes — considered as functions, symbols, and at the level of their Lie cobracket — and recount how the ‘nonclassical’ part of these amplitudes can be decomposed into specific functions evaluated on the A 2 or A 3 subalgebras of Gr( 4, n). We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the D 5 and A 5 subalgebras of Gr( 4, 7), and that these decompositions are themselves decomposable in terms of the same A 4 function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra.

Authors:
 [1];  [2]
  1. Univ. of Michigan, Ann Arbor, MI (United States). Leinweber Center for Theoretical Physics and Randall Lab. of Physics; Univ. of California, Santa Barbara, CA (United States). Kavli Inst. for Theoretical Physics
  2. Univ. of California, Santa Barbara, CA (United States). Kavli Inst. for Theoretical Physics; SLAC National Accelerator Lab., Menlo Park, CA (United States); Niels Bohr International Academy, Copenhagen (Denmark)
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1493350
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: 1; 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

Golden, John, and McLeod, Andrew J. Cluster algebras and the subalgebra constructibility of the seven-particle remainder function. United States: N. p., 2019. Web. doi:10.1007/jhep01(2019)017.
Golden, John, & McLeod, Andrew J. Cluster algebras and the subalgebra constructibility of the seven-particle remainder function. United States. doi:10.1007/jhep01(2019)017.
Golden, John, and McLeod, Andrew J. Thu . "Cluster algebras and the subalgebra constructibility of the seven-particle remainder function". United States. doi:10.1007/jhep01(2019)017. https://www.osti.gov/servlets/purl/1493350.
@article{osti_1493350,
title = {Cluster algebras and the subalgebra constructibility of the seven-particle remainder function},
author = {Golden, John and McLeod, Andrew J.},
abstractNote = {We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar N=4 supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory’s two-loop MHV amplitudes — considered as functions, symbols, and at the level of their Lie cobracket — and recount how the ‘nonclassical’ part of these amplitudes can be decomposed into specific functions evaluated on the A2 or A3 subalgebras of Gr(4, n). We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the D5 and A5 subalgebras of Gr(4, 7), and that these decompositions are themselves decomposable in terms of the same A4 function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra.},
doi = {10.1007/jhep01(2019)017},
journal = {Journal of High Energy Physics (Online)},
number = 1,
volume = 2019,
place = {United States},
year = {2019},
month = {1}
}

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

Citation Metrics:
Cited by: 6 works
Citation information provided by
Web of Science

Figures / Tables:

Figure 1 Figure 1: All possible triangulations of the pentagon. Mutating on the red chord moves you clockwise around the figure.

Save / Share:

Works referenced in this record:

Heptagons from the Steinmann cluster bootstrap
journal, February 2017

  • Dixon, Lance J.; Drummond, James; Harrington, Thomas
  • Journal of High Energy Physics, Vol. 2017, Issue 2
  • DOI: 10.1007/JHEP02(2017)137

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


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

Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond
journal, October 2005


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
  • DOI: 10.1007/JHEP12(2013)049

A simple collinear limit of scattering amplitudes at strong coupling
journal, March 2011


Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
journal, December 2016


Rationalizing loop integration
journal, August 2018

  • Bourjaily, Jacob L.; McLeod, Andrew J.; von Hippel, Matt
  • Journal of High Energy Physics, Vol. 2018, Issue 8
  • DOI: 10.1007/JHEP08(2018)184

Scattering amplitudes at strong coupling for 4K gluons
journal, December 2010


Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms
journal, March 2018


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


The double pentaladder integral to all orders
journal, July 2018

  • Caron-Huot, Simon; Dixon, Lance J.; von Hippel, Matt
  • Journal of High Energy Physics, Vol. 2018, Issue 7
  • DOI: 10.1007/JHEP07(2018)170

Local integrand representations of all two-loop amplitudes in planar SYM
journal, August 2015

  • Bourjaily, Jacob L.; Trnka, Jaroslav
  • Journal of High Energy Physics, Vol. 2015, Issue 8
  • DOI: 10.1007/JHEP08(2015)119

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
journal, August 2012


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


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

An analytic result for the two-loop seven-point MHV amplitude in N $$ \mathcal{N} $$ = 4 SYM
journal, August 2014


Iterated path integrals
journal, January 1977


Four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory
journal, April 2007


On-shell structures of MHV amplitudes beyond the planar limit
journal, June 2015

  • Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy
  • Journal of High Energy Physics, Vol. 2015, Issue 6
  • DOI: 10.1007/JHEP06(2015)179

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
  • DOI: 10.1007/JHEP01(2016)053

Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
journal, August 2018

  • Broedel, Johannes; Duhr, Claude; Dulat, Falko
  • Journal of High Energy Physics, Vol. 2018, Issue 8
  • DOI: 10.1007/JHEP08(2018)014

Cluster algebras II: Finite type classification
journal, May 2003


Cluster polylogarithms for scattering amplitudes
journal, November 2014

  • Golden, John; Paulos, Miguel F.; Spradlin, Marcus
  • Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue 47
  • DOI: 10.1088/1751-8113/47/47/474005

Multi-loop positivity of the planar N $$ \mathcal{N} $$ = 4 SYM six-point amplitude
journal, February 2017

  • Dixon, Lance J.; von Hippel, Matt; McLeod, Andrew J.
  • Journal of High Energy Physics, Vol. 2017, Issue 2
  • DOI: 10.1007/JHEP02(2017)112

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


All-helicity symbol alphabets from unwound amplituhedra
journal, May 2018

  • Prlina, I.; Spradlin, M.; Stankowicz, J.
  • Journal of High Energy Physics, Vol. 2018, Issue 5
  • DOI: 10.1007/JHEP05(2018)159

Cluster functions and scattering amplitudes for six and seven points
journal, July 2017

  • Harrington, Thomas; Spradlin, Marcus
  • Journal of High Energy Physics, Vol. 2017, Issue 7
  • DOI: 10.1007/JHEP07(2017)016

Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
journal, February 2010


New differential equations for on-shell loop integrals
journal, April 2011

  • Drummond, James M.; Henn, Johannes M.; Trnka, Jaroslav
  • Journal of High Energy Physics, Vol. 2011, Issue 4
  • DOI: 10.1007/JHEP04(2011)083

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
  • DOI: 10.1007/JHEP01(2014)091

Cluster Adjacency Properties of Scattering Amplitudes in N = 4 Supersymmetric Yang-Mills Theory
journal, April 2018


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


One-loop n-point gauge theory amplitudes, unitarity and collinear limits
journal, August 1994


Two-loop six-gluon maximally helicity violating amplitude in maximally supersymmetric Yang-Mills theory
journal, August 2008


Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
journal, December 2017

  • Abreu, Samuel; Britto, Ruth; Duhr, Claude
  • Journal of High Energy Physics, Vol. 2017, Issue 12
  • DOI: 10.1007/JHEP12(2017)090

Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms
journal, August 2018


Grassmannians and Cluster Algebras
journal, February 2006


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

Hedgehog bases for A n cluster polylogarithms and an application to six-point amplitudes
journal, November 2015

  • Parker, Daniel E.; Scherlis, Adam; Spradlin, Marcus
  • Journal of High Energy Physics, Vol. 2015, Issue 11
  • DOI: 10.1007/JHEP11(2015)136

Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010


Uniqueness of two-loop master contours
journal, October 2012

  • Caron-Huot, Simon; Larsen, Kasper J.
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)026

From polygons and symbols to polylogarithmic functions
journal, October 2012

  • Duhr, Claude; Gangl, Herbert; Rhodes, John R.
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)075

Maximally supersymmetric planar Yang-Mills amplitudes at five loops
journal, December 2007


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


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

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

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


Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction
journal, July 2017


Prescriptive unitarity
journal, June 2017

  • Bourjaily, Jacob L.; Herrmann, Enrico; Trnka, Jaroslav
  • Journal of High Energy Physics, Vol. 2017, Issue 6
  • DOI: 10.1007/JHEP06(2017)059

Cluster ensembles, quantization and the dilogarithm
journal, January 2009

  • Fock, Vladimir V.; Goncharov, Alexander B.
  • Annales scientifiques de l'École normale supérieure, Vol. 42, Issue 6
  • DOI: 10.24033/asens.2112

Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism
journal, May 2018

  • Broedel, Johannes; Duhr, Claude; Dulat, Falko
  • Journal of High Energy Physics, Vol. 2018, Issue 5
  • DOI: 10.1007/JHEP05(2018)093

    Works referencing / citing this record:

    Six-Gluon amplitudes in planar N$$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops
    journal, August 2019

    • Caron-Huot, Simon; Dixon, Lance J.; Dulat, Falko
    • Journal of High Energy Physics, Vol. 2019, Issue 8
    • DOI: 10.1007/jhep08(2019)016