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Scattering forms, worldsheet forms and amplitudes from subspaces

  • Song He
  • Gongwang Yan
  • Chi Zhang
  • Yong Zhang
Open Access
Regular Article - Theoretical Physics
  • 24 Downloads

Abstract

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.

Keywords

Differential and Algebraic Geometry Scattering Amplitudes 

Notes

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.

References

  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].ADSCrossRefMATHGoogle Scholar
  2. [2]
    N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].ADSCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle 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].ADSCrossRefMATHGoogle Scholar
  8. [8]
    N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Y.-J. Du and F. Teng, BCJ numerators from reduced Pfaffian, JHEP 04 (2017) 033 [arXiv:1703.05717] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle 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].MathSciNetCrossRefMATHGoogle 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].MathSciNetCrossRefMATHGoogle 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].ADSCrossRefMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle 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].ADSMathSciNetMATHGoogle 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].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Song He
    • 1
    • 2
  • Gongwang Yan
    • 1
    • 3
  • Chi Zhang
    • 1
    • 2
  • 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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