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

  • John Golden
  • Andrew J. McLeodEmail author
Open Access
Regular Article - Theoretical Physics


We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar \( \mathcal{N}=4 \) 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.


Scattering Amplitudes Supersymmetric Gauge Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016).Google Scholar
  2. [2]
    J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    J.L. Bourjaily, Positroids, Plabic Graphs and Scattering Amplitudes in Mathematica, arXiv:1212.6974 [INSPIRE].
  4. [4]
    L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar \( \mathcal{N}=4 \) super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
  6. [6]
    J.M. Drummond, G. Papathanasiou and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    L.J. Dixon, J. Drummond, T. Harrington, A.J. McLeod, G. Papathanasiou and M. Spradlin, Heptagons from the Steinmann Cluster Bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar \( \mathcal{N}=4 \) super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].
  10. [10]
    J. Golden and M. Spradlin, An analytic result for the two-loop seven-point MHV amplitude in \( \mathcal{N}=4 \) SYM, JHEP 08 (2014) 154 [arXiv:1406.2055] [INSPIRE].
  11. [11]
    J. Golden and M. Spradlin, The differential of all two-loop MHV amplitudes in \( \mathcal{N}= 4 \) Yang-Mills theory, JHEP 09 (2013) 111 [arXiv:1306.1833] [INSPIRE].
  12. [12]
    J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster Polylogarithms for Scattering Amplitudes, J. Phys. A 47 (2014) 474005 [arXiv:1401.6446] [INSPIRE].
  13. [13]
    J. Golden and M. Spradlin, A Cluster Bootstrap for Two-Loop MHV Amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    J. Drummond, J. Foster and Ö. Gürdoğan, Cluster Adjacency Properties of Scattering Amplitudes in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 120 (2018) 161601 [arXiv:1710.10953] [INSPIRE].
  15. [15]
    S. Caron-Huot and S. He, Jumpstarting the All-Loop S-matrix of Planar \( \mathcal{N}=4 \) Super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].
  16. [16]
    L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].CrossRefGoogle Scholar
  18. [18]
    L.J. Dixon, M. von Hippel, A.J. McLeod and J. Trnka, Multi-loop positivity of the planar \( \mathcal{N}=4 \) SYM six-point amplitude, JHEP 02 (2017) 112 [arXiv:1611.08325] [INSPIRE].
  19. [19]
    J. Drummond, J. Foster and Ö. Gürdoğan, Cluster adjacency beyond MHV, arXiv:1810.08149 [INSPIRE].
  20. [20]
    J.M. Drummond, J.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.L. Bourjaily, A.J. McLeod, M. von Hippel and M. Wilhelm, Rationalizing Loop Integration, JHEP 08 (2018) 184 [arXiv:1805.10281] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. Henn, E. Herrmann and J. Parra-Martinez, Bootstrapping two-loop Feynman integrals for planar \( \mathcal{N}=4 \) SYM, JHEP 10 (2018) 059 [arXiv:1806.06072] [INSPIRE].
  23. [23]
    S. Caron-Huot, L.J. Dixon, M. von Hippel, A.J. McLeod and G. Papathanasiou, The Double Pentaladder Integral to All Orders, JHEP 07 (2018) 170 [arXiv:1806.01361] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    I. Prlina, M. Spradlin, J. Stankowicz, S. Stanojevic and A. Volovich, All-Helicity Symbol Alphabets from Unwound Amplituhedra, JHEP 05 (2018) 159 [arXiv:1711.11507] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M.F. Paulos, M. Spradlin and A. Volovich, Mellin Amplitudes for Dual Conformal Integrals, JHEP 08 (2012) 072 [arXiv:1203.6362] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    J.L. Bourjaily and J. Trnka, Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM, JHEP 08 (2015) 119 [arXiv:1505.05886] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J.L. Bourjaily, E. Herrmann and J. Trnka, Prescriptive Unitarity, JHEP 06 (2017) 059 [arXiv:1704.05460] [INSPIRE].
  29. [29]
    J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    J.L. Bourjaily, Y.-H. He, A.J. Mcleod, M. Von Hippel and M. Wilhelm, Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms, Phys. Rev. Lett. 121 (2018) 071603 [arXiv:1805.09326] [INSPIRE].
  31. [31]
    J.L. Bourjaily, A.J. McLeod, M. von Hippel and M. Wilhelm, A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries, arXiv:1810.07689 [INSPIRE].
  32. [32]
    T. Harrington and M. Spradlin, Cluster Functions and Scattering Amplitudes for Six and Seven Points, JHEP 07 (2017) 016 [arXiv:1512.07910] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Am. Math. Soc. 15 (2002) 497 [math/0104151].
  34. [34]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Annales Sci. Ecole Norm. Sup. 42 (2009) 865 [math/0311245].
  36. [36]
    M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003) 899, [math/0208033].
  37. [37]
    J. Golden and A.J. McLeod, in progress.Google Scholar
  38. [38]
    C. Vergu, Polylogarithm identities, cluster algebras and the N = 4 supersymmetric theory, 2015, arXiv:1512.08113 [INSPIRE].
  39. [39]
    J.S. Scott, Grassmannians and cluster algebras, Proc. Lond. Math. Soc. 92 (2006) 345 [math/0311148].
  40. [40]
    L.J. Dixon, A brief introduction to modern amplitude methods, in Proceedings, 2012 European School of High-Energy Physics (ESHEP 2012), La Pommeraye, Anjou, France, June 06-19, 2012, pp. 31-67 (2014) [DOI:] [arXiv:1310.5353] [INSPIRE].
  41. [41]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
  42. [42]
    Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].
  43. [43]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].
  44. [44]
    Z. Bern et al., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [INSPIRE].
  45. [45]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [INSPIRE].
  46. [46]
    J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
  48. [48]
    Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].
  49. [49]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
  51. [51]
    S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003) 63 [math/0208229].
  52. [52]
    D. Parker, A. Scherlis, M. Spradlin and A. Volovich, Hedgehog bases for A n cluster polylogarithms and an application to six-point amplitudes, JHEP 11 (2015) 136 [arXiv:1507.01950] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    M. Sherman-Bennett, Combinatorics of χ -variables in finite type cluster algebras, arXiv:1803.02492.
  54. [54]
    W. Chang and B. Zhu, Cluster automorphism groups of cluster algebras of finite type, arXiv:1506.01950.
  55. [55]
    K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    F.C. Brown, Multiple zeta values and periods of moduli spaces \( {\overline{\mathfrak{M}}}_{0,n}\left(\mathbb{R}\right) \), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419].
  57. [57]
    A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
  58. [58]
    S. Caron-Huot, L.J. Dixon, M. von Hippel, A.J. McLeod and G. Papathanasiou, in progress.Google Scholar
  59. [59]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    C. Duhr, Mathematical aspects of scattering amplitudes, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders (TASI 2014), Boulder, Colorado, June 2-27, 2014, pp. 419-476, (2015) [DOI:] [arXiv:1411.7538] [INSPIRE].
  61. [61]
    H. Gangl, Multiple polylogarithms in weight 4, arXiv:1609.05557.
  62. [62]
    F. Brown, Mixed Tate motives over ℤ, arXiv:1102.1312.
  63. [63]
    F. Brown, Feynman amplitudes, coaction principle and cosmic Galois group, Commun. Num. Theor. Phys. 11 (2017) 453 [arXiv:1512.06409] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059.
  65. [65]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
  67. [67]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    S.J. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, American Mathematical Society (2000).Google Scholar
  69. [69]
    A. Suslin, K3 of a field and the bloch group, Proc. Stekov Inst. Math. 183 (1990) 217.zbMATHGoogle Scholar
  70. [70]
    A.B. Goncharov, Polylogarithms and motivic Galois groups, in Motives, Proc. Symp. Pure Math. 55 (1991) 43, Seattle, WA, American Mathematical Society, Providence, RI (1994).Google Scholar
  71. [71]
    N. Dan, Sur la conjecture de Zagier pour n = 4, arXiv:0809.3984.
  72. [72]
    H. Gangl, The Grassmannian complex and Goncharov’s motivic complex in weight 4, arXiv:1801.07816.
  73. [73]
    A.B. Goncharov and D. Rudenko, Motivic correlators, cluster varieties and Zagier’s conjecture on ζ F (4), arXiv:1803.08585.
  74. [74]
    L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    G. Yang, A simple collinear limit of scattering amplitudes at strong coupling, JHEP 03 (2011) 087 [arXiv:1006.3306] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  76. [76]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
  77. [77]
    O. Steinmann, Über den Zusammenhang zwischen den Wightmanfunktionen und der retardierten Kommutatoren, Helv. Physica Acta 33 (1960) 257.MathSciNetzbMATHGoogle Scholar
  78. [78]
    O. Steinmann, Wightman-Funktionen und retardierten Kommutatoren. II, Helv. Physica Acta 33 (1960) 347.Google Scholar
  79. [79]
    K.E. Cahill and H.P. Stapp, Optical Theorems and Steinmann Relations, Annals Phys. 90 (1975) 438 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  80. [80]
    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
  81. [81]
    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, arXiv:1809.10698 [INSPIRE].
  82. [82]
    G. Yang, Scattering amplitudes at strong coupling for 4K gluons, JHEP 12 (2010) 082 [arXiv:1004.3983] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
  84. [84]
    F. Brown, Notes on Motivic Periods, arXiv:1512.06410.
  85. [85]
    S. Abreu, R. Britto, C. Duhr and E. Gardi, Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction, Phys. Rev. Lett. 119 (2017) 051601 [arXiv:1703.05064] [INSPIRE].
  86. [86]
    S. Abreu, R. Britto, C. Duhr and E. Gardi, Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case, JHEP 12 (2017) 090 [arXiv:1704.07931] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, JHEP 08 (2018) 014 [arXiv:1803.10256] [INSPIRE].ADSCrossRefGoogle Scholar
  88. [88]
    F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Leinweber Center for Theoretical Physics and Randall Laboratory of Physics, Department of PhysicsUniversity of MichiganAnn ArborU.S.A.
  2. 2.Kavli Institute for Theoretical PhysicsUC Santa BarbaraSanta BarbaraU.S.A.
  3. 3.SLAC National Accelerator LaboratoryStanford UniversityStanfordU.S.A.
  4. 4.Niels Bohr International AcademyCopenhagenDenmark

Personalised recommendations