Journal of High Energy Physics

, 2019:271 | Cite as

The positive geometry for 𝜙p interactions

  • Prashanth RamanEmail author
Open Access
Regular Article - Theoretical Physics


Starting with the seminal work of Arkani-Hamed et al. [1], in [2], the “Ampli- tuhedron program” was extended to analyzing (planar) amplitudes in massless 𝜙 4 theory. In this paper we show that the program can be further extended to include 𝜙p (p > 4) interactions. We show that tree-level planar amplitudes in these theories can be obtained from geometry of polytopes called accordiohedron which naturally sits inside kinematic space. As in the case of quartic interactions the accordiohedron of a given dimension is not unique, and we show that a weighted sum of residues of the canonical form on these polytopes can be used to compute scattering amplitudes. We finally provide a prescription to compute the weights and demonstrate how it works in various examples.


Scattering Amplitudes Differential and Algebraic Geometry 


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, Y. Bai, S. He and G. Yan, Scattering forms and the positive geometry of kinematics, color and the worldsheet, JHEP05 (2018) 096 [arXiv:1711.09102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    P. Banerjee, A. Laddha and P. Raman, Stokes polytopes: the positive geometry for 𝜙4interactions, JHEP08 (2019) 067 [arXiv:1811.05904] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP12 (2014) 182 [arXiv:1312.7878] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    N. Arkani-Hamed, A. Hodges and J. Trnka, Positive amplitudes in the Amplituhedron, JHEP08 (2015) 030 [arXiv:1412.8478] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in binary, JHEP01 (2018) 016 [arXiv:1704.05069] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    N. Arkani-Hamed et al., Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridghe U.K. (2016), arXiv:1212.5605 [INSPIRE].
  7. [7]
    N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP10 (2014) 030 [arXiv:1312.2007] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    N. Arkani-Hamed, Y. Bai and T. Lam, Positive geometries and canonical forms, JHEP11 (2017) 039 [arXiv:1703.04541] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    N. Arkani-Hamed, C. Langer, A. Yelleshpur Srikant and J. Trnka, Deep into the Amplituhedron: amplitude singularities at all loops and legs, Phys. Rev. Lett.122 (2019) 051601 [arXiv:1810.08208] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    G. Salvatori and S.L. Cacciatori, Hyperbolic geometry and amplituhedra in 1 + 2 dimensions, JHEP08 (2018) 167 [arXiv:1803.05809] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    G. Salvatori, 1-loop amplitudes from the halohedron, arXiv:1806.01842 [INSPIRE].
  12. [12]
    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
  13. [13]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Scattering equations and Feynman diagrams, JHEP09 (2015) 136 [arXiv:1507.00997] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, String-like dual models for scalar theories, JHEP12 (2016) 019 [arXiv:1610.04228] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    T. Manneville and V. Pilaud, Geometric realizations of the accordion complex of a dissection, Discrete Comput. Geom.61 (2019) 507.MathSciNetCrossRefGoogle Scholar
  16. [16]
    A. Garver and T. Mcconville, Oriented flip graphs, non-crossing tree partitions, and representation theory of tiling algebras, Glasgow Math. J. (2019) [arXiv:1604.06009].
  17. [17]
    S.L. Devadoss, A realization of graph associahedra, Discr. Math.309 (2009) 271.MathSciNetCrossRefGoogle Scholar
  18. [18]
    S. He, G. Yan, C. Zhang and Y. Zhang, Scattering forms, worldsheet forms and amplitudes from subspaces, JHEP08 (2018) 040 [arXiv:1803.11302] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Theor.A 49 (1988) 381.MathSciNetCrossRefGoogle Scholar
  20. [20]
    R.M.L. Blind, Puzzles and polytope isomorphisms, Aequat. Math.34 (1987) 287.Google Scholar
  21. [21]
    Y. Baryshnikov, On Stokes sets, New Develop. Singul. Theor.21 (2001) 65.MathSciNetCrossRefGoogle Scholar
  22. [22]
    F. Harary, E.M. Palmer and R.C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math.11 (1975) 371.MathSciNetCrossRefGoogle Scholar
  23. [23]
    N. Apostolakis, Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections, arXiv:1807.11602.
  24. [24]
    M. Jagadale, N. Kalyanapuram and A.P. Balakrishnan, Accordiohedra as positive geometries for generic scalar field theories, arXiv:1906.12148 [INSPIRE].
  25. [25]
    C. Cheung, TASI lectures on scattering amplitudes, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics (TASI 2016), June 6–July 1, Boulder, U.S.A. (2016), arXiv:1708.03872, [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesChennaiIndia
  2. 2.Homi Bhabha National InstituteMumbaiIndia

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