Journal of High Energy Physics

, 2016:178 | Cite as

Comments on the evaluation of massless scattering

Open Access
Regular Article - Theoretical Physics


The goal of this work is threefold. First, we give an expression of the most general five point integral on \( {\mathrm{\mathcal{M}}}_{0,n} \) in terms of Chebyshev polynomials. Second, we choose a special kinematics that transforms the polynomial form of the scattering equations to a linear system of symmetric polynomials. We then explain how this can be used to explicitly evaluate arbitrary point integrals on \( {\mathrm{\mathcal{M}}}_{0,n} \). Third, we comment on the recently presented method of companion matrices and we show its equivalence to the elimination theory and an algorithm previously developed by one of the authors.


Scattering Amplitudes 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]
    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
  2. [2]
    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].CrossRefADSGoogle Scholar
  3. [3]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].CrossRefADSGoogle Scholar
  4. [4]
    F. Cachazo, S. He and E.Y. Yuan, Einstein- Yang-Mills scattering amplitudes from scattering equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    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].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    L. Dolan and P. Goddard, Proof of the formula of Cachazo, He and Yuan for Yang-Mills tree amplitudes in arbitrary dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    D. Fairlie and D. Roberts, Dual models without tachyons — A new approach, unpublished Durham preprint PRINT-72-2440 (1972).Google Scholar
  8. [8]
    D. Roberts, Mathematical structure of dual amplitudes, Ph.D. thesis, Durham University, Durham, U.K. (1972).
  9. [9]
    D.B. Fairlie, A coding of real null four-momenta into world-sheet co-ordinates, Adv. Math. Phys. 2009 (2009) 284689 [arXiv:0805.2263] [INSPIRE].CrossRefMathSciNetMATHGoogle Scholar
  10. [10]
    D.J. Gross and P.F. Mende, String theory beyond the Planck scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    C. Kalousios, Scattering equations, generating functions and all massless five point tree amplitudes, JHEP 05 (2015) 054 [arXiv:1502.07711] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    L. Dolan and P. Goddard, The polynomial form of the scattering equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    F. Cachazo and H. Gomez, Computation of contour integrals on \( {\mathrm{\mathcal{M}}}_{0,n} \), arXiv:1505.03571 [INSPIRE].
  14. [14]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Integration rules for scattering equations, JHEP 09 (2015) 129 [arXiv:1506.06137] [INSPIRE].CrossRefADSGoogle Scholar
  15. [15]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Scattering equations and Feynman diagrams, JHEP 09 (2015) 136 [arXiv:1507.00997] [INSPIRE].CrossRefADSGoogle Scholar
  16. [16]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, P. Tourkine and P. Vanhove, Scattering equations and string theory amplitudes, Phys. Rev. D 90 (2014) 106002 [arXiv:1403.4553] [INSPIRE].ADSGoogle Scholar
  17. [17]
    R. Huang, J. Rao, B. Feng and Y.-H. He, An algebraic approach to the scattering equations, JHEP 12 (2015) 056 [arXiv:1509.04483] [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    M. Sogaard and Y. Zhang, Scattering equations and global duality of residues, arXiv:1509.08897 [INSPIRE].
  19. [19]
    C. Kalousios, Massless scattering at special kinematics as Jacobi polynomials, J. Phys. A 47 (2014) 215402 [arXiv:1312.7743] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  20. [20]
    S. Weinzierl, On the solutions of the scattering equations, JHEP 04 (2014) 092 [arXiv:1402.2516] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    C.S. Lam, Permutation symmetry of the scattering equations, Phys. Rev. D 91 (2015) 045019 [arXiv:1410.8184] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    G. Dvali, C. Gomez, R.S. Isermann, D. Lüst and S. Stieberger, Black hole formation and classicalization in ultra-Planckian 2 → N scattering, Nucl. Phys. B 893 (2015) 187 [arXiv:1409.7405] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  23. [23]
    F. Cachazo, S. He and E.Y. Yuan, Scattering in three dimensions from rational maps, JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    B. Sturmfels, Solving systems of polynomial equations, in CBMS Regional Conference Series in Mathematics, American Mathematical Society, U.S.A. (2002).Google Scholar
  25. [25]
    R. Fröberg, An introduction to Gröbner bases, Wiley (1998).Google Scholar
  26. [26]
    B. Buchberger, Gröbner basis: An algorithmic method in polynomial ideal theory, in ultidimensional systems theory, N.K. Bose ed., Reidel (1985).Google Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Physics Division, National Center for Theoretical SciencesNational Tsing-Hua UniversityHsinchuRepublic of China
  2. 2.ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, UNESP-Universidade Estadual PaulistaSão PauloBrasil

Personalised recommendations