Abstract
We present an algorithm that leads to BCJ numerators satisfying manifestly the three properties proposed by Broedel and Carrasco in [42]. We explicitly calculate the numerators at 4, 5 and 6-points and show that the relabeling property is generically satisfied.
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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].
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].
J.J. Carrasco and H. Johansson, Five-point amplitudes in N = 4 super-Yang-Mills theory and N = 8 supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying multiloop integrands and ultraviolet divergences of gauge theory and gravity amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
R.H. Boels, B.A. Kniehl, O.V. Tarasov and G. Yang, Color-kinematic duality for form factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].
J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled N ≤ 4 supergravity, JHEP 03 (2013) 056 [arXiv:1212.1146] [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].
N.E.J. Bjerrum-Bohr, T. Dennen, R. Monteiro and D. O’Connell, Integrand oxidation and one-loop colour-dual numerators in N = 4 gauge theory, JHEP 07 (2013) 092 [arXiv:1303.2913] [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].
M. Chiodaroli, Q. Jin and R. Roiban, Color/kinematics duality for general Abelian orbifolds of N = 4 super Yang-Mills theory, JHEP 01 (2014) 152 [arXiv:1311.3600] [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, H. Johansson and T. Sondergaard, Monodromy-like relations for finite loop amplitudes, JHEP 05 (2011) 039 [arXiv:1103.6190] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, BCJ relation of color scalar theory and KLT relation of gauge theory, JHEP 08 (2011) 129 [arXiv:1105.3503] [INSPIRE].
R.H. Boels and R.S. Isermann, New relations for scattering amplitudes in Yang-Mills theory at loop level, Phys. Rev. D 85 (2012) 021701 [arXiv:1109.5888] [INSPIRE].
R.H. Boels and R.S. Isermann, Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts, JHEP 03 (2012) 051 [arXiv:1110.4462] [INSPIRE].
Y.-J. Du and H. Lüo, On general BCJ relation at one-loop level in Yang-Mills theory, JHEP 01 (2013) 129 [arXiv:1207.4549] [INSPIRE].
S. Oxburgh and C.D. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].
R. Saotome and R. Akhoury, Relationship between gravity and gauge scattering in the high energy limit, JHEP 01 (2013) 123 [arXiv:1210.8111] [INSPIRE].
J. Nohle, Color-kinematics duality in one-loop four-gluon amplitudes with matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
Z. Bern, C. Boucher-Veronneau and H. Johansson, N ≥ 4 supergravity amplitudes from gauge theory at one loop, Phys. Rev. D 84 (2011) 105035 [arXiv:1107.1935] [INSPIRE].
Z. Bern and T. Dennen, A color dual form for gauge-theory amplitudes, Phys. Rev. Lett. 107 (2011) 081601 [arXiv:1103.0312] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, An algebraic approach to BCJ numerators, JHEP 03 (2013) 050 [arXiv:1212.6168] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, The construction of dual-trace factor in Yang-Mills theory, JHEP 07 (2013) 057 [arXiv:1304.2978] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, Note on construction of dual-trace factor in Yang-Mills theory, JHEP 10 (2013) 069 [arXiv:1305.2996] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, Dual-color decompositions at one-loop level in Yang-Mills theory, JHEP 06 (2014) 157 [arXiv:1402.6805] [INSPIRE].
R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
H. Kawai, D.C. Lewellen and S.H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
S.H. Henry Tye and Y. Zhang, Dual identities inside the gluon and the graviton scattering amplitudes, JHEP 06 (2010) 071 [Erratum ibid. 04 (2011) 114] [arXiv:1003.1732] [INSPIRE].
Z. Bern, A. De Freitas and H.L. Wong, On the coupling of gravitons to matter, Phys. Rev. Lett. 84 (2000) 3531 [hep-th/9912033] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Gravity and Yang-Mills amplitude relations, Phys. Rev. D 82 (2010) 107702 [arXiv:1005.4367] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, New identities among gauge theory amplitudes, Phys. Lett. B 691 (2010) 268 [arXiv:1006.3214] [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].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
R. Monteiro and D. O’Connell, The kinematic algebra from the self-dual sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, R. Monteiro and D. O’Connell, Algebras for amplitudes, JHEP 06 (2012) 061 [arXiv:1203.0944] [INSPIRE].
R. Monteiro and D. O’Connell, The kinematic algebras from the scattering equations, JHEP 03 (2014) 110 [arXiv:1311.1151] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and KLT orthogonality, arXiv:1306.6575 [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimension, 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].
R.H. Boels and R.S. Isermann, On powercounting in perturbative quantum gravity theories through color-kinematic duality, JHEP 06 (2013) 017 [arXiv:1212.3473] [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].
Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The momentum kernel of gauge and gravity theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].
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ArXiv ePrint: 1403.6262
The unusual ordering of authors in this paper is just to fit outdated requirement of Chinese University for recognition of contributions (Chih-Hao Fu).
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Fu, CH., Du, YJ. & Feng, B. Note on symmetric BCJ numerator. J. High Energ. Phys. 2014, 98 (2014). https://doi.org/10.1007/JHEP08(2014)098
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DOI: https://doi.org/10.1007/JHEP08(2014)098