Abstract
Color-ordered tree level scattering amplitudes in Yang-Mills theories can be written as a sum over terms which display the various propagator poles of Feynman diagrams. The numerators in these expressions which are obtained by straightforward application of Feynman rules are not satisfying any particular relations, typically. However, by reshuffling terms, it is known that one can arrive at a set of numerators which satisfy the same Jacobi identity as the corresponding color factors. By extending previous work by us we show how this can be systematically accomplished within a Lagrangian framework. We construct an effective Lagrangian which yields tree-level color-kinematic symmetric numerators in Yang-Mills theories in a light-like gauge at five-points. The five-point effective Lagrangian is non-local and it is zero by Jacobi identity. The numerators obtained from it respect the original pole structure of the color-ordered amplitude. We discuss how this procedure can be systematically extended to higher order.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the Square of Gauge Theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Proof of Gravity and Yang-Mills Amplitude Relations, JHEP 09 (2010) 067 [arXiv:1007.3111] [INSPIRE].
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].
S.H. Henry Tye and Y. Zhang, Dual Identities inside the Gluon and the Graviton Scattering Amplitudes, JHEP 06 (2010) 071 [Erratum ibid. 1104 (2011) 114] [arXiv:1003.1732] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [INSPIRE].
R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
R.H. Boels, R.S. Isermann, R. Monteiro and D. O’Connell, Colour-Kinematics Duality for One-Loop Rational Amplitudes, JHEP 04 (2013) 107 [arXiv:1301.4165] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, Y.-t. Huang and J. Nohle, Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops, arXiv:1303.6605 [INSPIRE].
R. Kleiss and H. Kuijf, Multi-Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
D. Vaman and Y.-P. Yao, Constraints and generalized gauge transformations on tree-level gluon and graviton amplitudes, JHEP 11 (2010) 028 [arXiv:1007.3475] [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].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
J. Broedel and J.J.M. Carrasco, Virtuous Trees at Five and Six Points for Yang-Mills and Gravity, Phys. Rev. D 84 (2011) 085009 [arXiv:1107.4802] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, An algebraic approach to BCJ numerators, JHEP 03 (2013) 050 [arXiv:1212.6168] [INSPIRE].
S.G. Naculich, Scattering equations and virtuous kinematic numerators and dual-trace functions, JHEP 07 (2014) 143 [arXiv:1404.7141] [INSPIRE].
M. Tolotti and S. Weinzierl, Construction of an effective Yang-Mills Lagrangian with manifest BCJ duality, JHEP 07 (2013) 111 [arXiv:1306.2975] [INSPIRE].
G. Chalmers and W. Siegel, Simplifying algebra in Feynman graphs. Part 2. Spinor helicity from the space-cone, Phys. Rev. D 59 (1999) 045013 [hep-ph/9801220] [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].
D. Vaman and Y.-P. Yao, QCD recursion relations from the largest time equation, JHEP 04 (2006) 030 [hep-th/0512031] [INSPIRE].
P. Mansfield, The Lagrangian origin of MHV rules, JHEP 03 (2006) 037 [hep-th/0511264] [INSPIRE].
S. Mandelstam, Light Cone Superspace and the Ultraviolet Finiteness of the N = 4 Model, Nucl. Phys. B 213 (1983) 149 [INSPIRE].
D. Vaman and Y.-P. Yao, The Space-Cone Gauge, Lorentz invariance and on-shell recursion for one-loop Yang-Mills amplitudes, arXiv:0805.2645 [INSPIRE].
S. Ananth and S. Theisen, KLT relations from the Einstein-Hilbert Lagrangian, Phys. Lett. B 652 (2007) 128 [arXiv:0706.1778] [INSPIRE].
A. Karlberg and T. Sondergaard, Feynman rules for QCD in space-cone gauge, arXiv:1201.1441 [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: 1408.2818
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
Vaman, D., Yao, YP. Color kinematic symmetric (BCJ) numerators in a light-like gauge. J. High Energ. Phys. 2014, 36 (2014). https://doi.org/10.1007/JHEP12(2014)036
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2014)036