Abstract
We argue that one does not need to know the explicit solutions of the scattering equations in order to evaluate a given amplitude. We consider the most general quantity consistent with SL(2, ℂ) invariance that can appear in an amplitude that admits a scattering equation description. This quantity depends on all cross ratios that can be formed from n points and we evaluate it for the first non-trivial case of n = 5. The combinatorial nature of the problem is captured through the construction of an appropriate generating function that depends on five variables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
D.B. Fairlie and D.E. Roberts, Dual Models without Tachyons — A New Approach, PRINT-72-2440 (1972) [INSPIRE].
D.E. Roberts, Mathematical Structure of Dual Amplitudes, Ph.D. Thesis, Durham University, Durham U.K. (1972), p. 73 and online at http://etheses.dur.ac.uk/8662/.
D.B. Fairlie, A Coding of Real Null Four-Momenta into World-Sheet Coordinates, Adv. Math. Phys. 2009 (2009) 284689 [arXiv:0805.2263] [INSPIRE].
D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].
E. Witten, Parity invariance for strings in twistor space, Adv. Theor. Math. Phys. 8 (2004) 779 [hep-th/0403199] [INSPIRE].
P. Caputa and S. Hirano, Observations on Open and Closed String Scattering Amplitudes at High Energies, JHEP 02 (2012) 111 [arXiv:1108.2381] [INSPIRE].
P. Caputa, Lightlike contours with fermions, Phys. Lett. B 716 (2012) 475 [arXiv:1205.6369] [INSPIRE].
Y. Makeenko and P. Olesen, The QCD scattering amplitude from area behaved Wilson loops, Phys. Lett. B 709 (2012) 285 [arXiv:1111.5606] [INSPIRE].
F. Cachazo, Fundamental BCJ Relation in N = 4 SYM From The Connected Formulation, arXiv:1206.5970 [INSPIRE].
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].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
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].
S.G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, arXiv:1412.3479 [INSPIRE].
S. Weinzierl, Fermions and the scattering equations, JHEP 03 (2015) 141 [arXiv:1412.5993] [INSPIRE].
S.G. Naculich, CHY representations for gauge theory and gravity amplitudes with up to three massive particles, arXiv:1501.03500 [INSPIRE].
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].
L. Dolan and P. Goddard, The Polynomial Form of the Scattering Equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].
T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].
E. Casali and P. Tourkine, Infrared behaviour of the one-loop scattering equations and supergravity integrands, JHEP 04 (2015) 013 [arXiv:1412.3787] [INSPIRE].
T. Adamo and E. Casali, Scattering equations, supergravity integrands and pure spinors, arXiv:1502.06826 [INSPIRE].
L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].
N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].
Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor Strings in Four Dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].
C. Kalousios, Massless scattering at special kinematics as Jacobi polynomials, J. Phys. A 47 (2014) 215402 [arXiv:1312.7743] [INSPIRE].
S. Weinzierl, On the solutions of the scattering equations, JHEP 04 (2014) 092 [arXiv:1402.2516] [INSPIRE].
C.S. Lam, Permutation Symmetry of the Scattering Equations, Phys. Rev. D 91 (2015) 045019 [arXiv:1410.8184] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1502.07711
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kalousios, C. Scattering equations, generating functions and all massless five point tree amplitudes. J. High Energ. Phys. 2015, 54 (2015). https://doi.org/10.1007/JHEP05(2015)054
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2015)054