Scattering forms, worldsheet forms and amplitudes from subspaces

  • Song He
  • Gongwang Yan
  • Chi ZhangEmail author
  • Yong Zhang
Open Access
Regular Article - Theoretical Physics


We present a general construction of two types of differential forms, based on any (n−3)-dimensional subspace in the kinematic space of n massless particles. The first type is the so-called projective, scattering forms in kinematic space, while the second is defined in the moduli space of n-punctured Riemann spheres which we call worldsheet forms. We show that the pushforward of worldsheet forms, by summing over solutions of scattering equations, gives the corresponding scattering forms, which generalizes the results of [1]. The pullback of scattering forms to subspaces can have natural interpretations as amplitudes in terms of Bern-Carrasco-Johansson double-copy construction or Cachazo-He-Yuan formula. As an application of our formalism, we construct in this way a large class of d log scattering forms and worldsheet forms, which are in one-to-one correspondence with non-planar MHV leading singularities in \( \mathcal{N}=4 \) super-Yang-Mills. For every leading singularity function, we present a new determinant formula in moduli space, as well as a (combinatoric) polytope and associated scattering form in kinematic space. These include the so-called Cayley cases, where in each case the scattering form is the canonical forms of a convex polytope in the subspace, and scattering equations admit elegant rewritings as a map from the moduli space to the subspace.


Differential and Algebraic Geometry Scattering Amplitudes 


Open Access

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  1. [1]
    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].ADSCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    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
  6. [6]
    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].ADSGoogle Scholar
  7. [7]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  8. [8]
    N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    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].ADSMathSciNetGoogle Scholar
  11. [11]
    Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    X. Gao, S. He and Y. Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, JHEP 11 (2017) 144 [arXiv:1708.08701] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    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
  15. [15]
    F. Cachazo, Combinatorial Factorization, arXiv:1710.04558 [INSPIRE].
  16. [16]
    Y.-J. Du and C.-H. Fu, Explicit BCJ numerators of nonlinear simga model, JHEP 09 (2016) 174 [arXiv:1606.05846] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α-corrections from the open string, JHEP 06 (2017) 093 [arXiv:1608.02569] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Y.-J. Du and F. Teng, BCJ numerators from reduced Pfaffian, JHEP 04 (2017) 033 [arXiv:1703.05717] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Y.-J. Du and Y. Zhang, Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes, arXiv:1803.01701 [INSPIRE].
  20. [20]
    F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Carr and S.L. Devadoss, Coxeter Complexes and Graph-Associahedra, math/0407229.
  22. [22]
    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), [arXiv:1212.5605] [INSPIRE].
  23. [23]
    A. Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. 2009 (2009) 1026, [math/0507163].
  24. [24]
    R. Stanley, Enumerative Combinatorics, vol. 1, Cambridge University Press, (1996).Google Scholar
  25. [25]
    M.C. Abbati and A. Mania, On the spectrum of holonomy algebras, J. Geom. Phys. 44 (2002) 96 [math-ph/0202004] [INSPIRE].
  26. [26]
    S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    N. Early, Canonical Bases for Permutohedral Plates, arXiv:1712.08520 [INSPIRE].
  28. [28]
    G. Salvatori and S.L. Cacciatori, Hyperbolic Geometry and Amplituhedra in 1+2 dimensions, arXiv:1803.05809 [INSPIRE].
  29. [29]
    L. de la Cruz, A. Kniss and S. Weinzierl, Properties of scattering forms and their relation to associahedra, JHEP 03 (2018) 064 [arXiv:1711.07942] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M.A.C. Torres, Cluster Algebras in Kinematic Space of Scattering Amplitudes, arXiv:1712.06161 [INSPIRE].
  31. [31]
    H. Frost, Biadjoint scalar tree amplitudes and intersecting dual associahedra, JHEP 06 (2018) 153 [arXiv:1802.03384] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    N. Early, Generalized permutohedra in the kinematic space, arXiv:1804.05460 [INSPIRE].
  33. [33]
    E. Casali and P. Tourkine, Infrared behaviour of the one-loop scattering equations and supergravity integrands, JHEP 04 (2015) 013 [arXiv:1412.3787] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    S. He and E.Y. Yuan, One-loop Scattering Equations and Amplitudes from Forward Limit, Phys. Rev. D 92 (2015) 105004 [arXiv:1508.06027] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  37. [37]
    F. Cachazo, S. He and E.Y. Yuan, One-Loop Corrections from Higher Dimensional Tree Amplitudes, JHEP 08 (2016) 008 [arXiv:1512.05001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-Loop Scattering Amplitudes from the Riemann Sphere, Phys. Rev. D 94 (2016) 125029 [arXiv:1607.08887] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  39. [39]
    S. He and O. Schlotterer, New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level, Phys. Rev. Lett. 118 (2017) 161601 [arXiv:1612.00417] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    S. He, O. Schlotterer and Y. Zhang, New BCJ representations for one-loop amplitudes in gauge theories and gravity, Nucl. Phys. B 930 (2018) 328 [arXiv:1706.00640] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    H. Gomez, S. Mizera and G. Zhang, CHY Loop Integrands from Holomorphic Forms, JHEP 03 (2017) 092 [arXiv:1612.06854] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Song He
    • 1
    • 2
  • Gongwang Yan
    • 1
    • 3
  • Chi Zhang
    • 1
    • 2
    Email author
  • Yong Zhang
    • 1
    • 2
  1. 1.CAS Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingChina
  2. 2.University of Chinese Academy of SciencesBeijingChina
  3. 3.Institute for Advanced StudyTsinghua UniversityBeijingChina

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