Δ-algebra and scattering amplitudes

  • Freddy Cachazo
  • Nick Early
  • Alfredo Guevara
  • Sebastian MizeraEmail author
Open Access
Regular Article - Theoretical Physics


In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of G(2, n) as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the Δ-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the Δ-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to G(k, n).


Scattering Amplitudes Supersymmetric Gauge Theory Field Theories in Higher Dimensions 


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, Cambridge U.K. (2016) [arXiv:1212.5605] [INSPIRE].
  2. [2]
    A. Postnikov, Total positivity, Grassmannians, and networks, math/0609764.
  3. [3]
    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
  4. [4]
    S. He and C. Zhang, Notes on Scattering Amplitudes as Differential Forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
  6. [6]
    E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [INSPIRE].
  7. [7]
    F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE].
  8. [8]
    N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
  10. [10]
    N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    L. Ferro, T. Lukowski and M. Parisi, Amplituhedron meets Jeffrey-Kirwan Residue, J. Phys. A 52 (2019) 045201 [arXiv:1805.01301] [INSPIRE].
  12. [12]
    N. Early and V. Reiner, On configuration spaces and Whitehouse’s lifts of the Eulerian representations, arXiv:1808.04007.
  13. [13]
    D. Moseley, N. Proudfoot and B. Young, The Orlik-Terao algebra and the cohomology of configuration space, Exp. Math. 26 (2017) 373 [arXiv:1603.01189].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    B. Knudsen, Configuration spaces in algebraic topology, arXiv:1803.11165.
  15. [15]
    V.I. Arnol’d, The cohomology ring of the colored braid group, Math. Notes Acad. Sci. USSR 5 (1969) 138.Google Scholar
  16. [16]
    W. Fulton and R. MacPherson, A compactification of configuration spaces, Annals Math. 139 (1994) 183 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    B. Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996) 1057.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    I. Kriz, On the rational homotopy type of configuration spaces, Ann. Math. 139 (1994) 227.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Ocneanu, Higher Representation Theory in Math and Physics, Harvard University course PHYSICS 267 (2017),
  20. [20]
    N. Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [INSPIRE].
  21. [21]
    N. Early, Generalized Permutohedra, Scattering Amplitudes and a Cubic Three-Fold, arXiv:1709.03686 [INSPIRE].
  22. [22]
    M. Enciso, Volumes of Polytopes Without Triangulations, JHEP 10 (2017) 071 [arXiv:1408.0932] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Enciso, Logarithms and Volumes of Polytopes, JHEP 04 (2018) 016 [arXiv:1612.07370] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    B. Feng and M. Luo, An Introduction to On-shell Recursion Relations, Front. Phys. (Beijing) 7 (2012) 533 [arXiv:1111.5759] [INSPIRE].CrossRefGoogle Scholar
  26. [26]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
  27. [27]
    Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
  28. [28]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
  30. [30]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. Franco, D. Galloni, B. Penante and C. Wen, Non-Planar On-Shell Diagrams, JHEP 06 (2015) 199 [arXiv:1502.02034] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    R. Frassek and D. Meidinger, Yangian-type symmetries of non-planar leading singularities, JHEP 05 (2016) 110 [arXiv:1603.00088] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    J.L. Bourjaily, S. Franco, D. Galloni and C. Wen, Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams, JHEP 10 (2016) 003 [arXiv:1607.01781] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    P. Deligne and J.W. Morgan, Notes on Supersymmetry (following Joseph Bernstein), in Quantum Fields and Strings: A Course for Mathematicians, AMS Press, New York U.S.A. (1999), pg. 41.Google Scholar
  37. [37]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].
  40. [40]
    S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    R. Kleiss and H. Kuijf, Multi-Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
  42. [42]
    L.J. Dixon, Calculating scattering amplitudes efficiently, in QCD and beyond. Proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder U.S.A. (1995), pg. 539 [hep-ph/9601359] [INSPIRE].
  43. [43]
    F. Cachazo, Fundamental BCJ Relation in N = 4 SYM From The Connected Formulation, arXiv:1206.5970 [INSPIRE].
  44. [44]
    P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg Germany (2013).Google Scholar
  45. [45]
    H. Esnault, V. Schechtman and E. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992) 557.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    S. He and Y. Zhang, New Formulas for Amplitudes from Higher-Dimensional Operators, JHEP 02 (2017) 019 [arXiv:1608.08448] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    S. He, G. Yan, C. Zhang and Y. Zhang, Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces, JHEP 08 (2018) 040 [arXiv:1803.11302] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  49. [49]
    U. Pachner, P.l. homeomorphic manifolds are equivalent by elementary shellings, Eur. J. Combin. 12 (1991) 129.Google Scholar
  50. [50]
    N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, JHEP 05 (2018) 096 [arXiv:1711.09102] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    S. He and Q. Yang, An Etude on Recursion Relations and Triangulations, arXiv:1810.08508 [INSPIRE].
  52. [52]
    P. Benincasa, On-shell diagrammatics and the perturbative structure of planar gauge theories, arXiv:1510.03642 [INSPIRE].
  53. [53]
    P. Heslop and A.E. Lipstein, On-shell diagrams for \( \mathcal{N}=8 \) supergravity amplitudes, JHEP 06 (2016) 069 [arXiv:1604.03046] [INSPIRE].
  54. [54]
    E. Herrmann and J. Trnka, Gravity On-shell Diagrams, JHEP 11 (2016) 136 [arXiv:1604.03479] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    N. Arkani-Hamed, A. Hodges and J. Trnka, Positive Amplitudes In The Amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
  57. [57]
    R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    V. Reiner, Lectures on matroids and oriented matroids,
  59. [59]
    F. Cachazo and Y. Geyer, A ’Twistor String’ Inspired Formula For Tree-Level Scattering Amplitudes in N = 8 SUGRA, arXiv:1206.6511 [INSPIRE].
  60. [60]
    F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Massachusetts Institute of TechnologyCambridgeUnited States
  3. 3.Department of Physics & AstronomyUniversity of WaterlooWaterlooCanada
  4. 4.CECs Valdivia & Departamento de FísicaUniversidad de ConcepciónConcepciónChile

Personalised recommendations