An algebraic approach to BCJ numerators

  • Chih-Hao Fu
  • Yi-Jian Du
  • Bo Feng


One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.


Scattering Amplitudes Gauge Symmetry 


  1. [1]
    Z. Bern, J. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].MathSciNetADSGoogle Scholar
  2. [2]
    N. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
  4. [4]
    N. 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].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    B. Feng, R. Huang and Y. Jia, Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].MathSciNetADSGoogle Scholar
  7. [7]
    Y. Jia, R. Huang and C.-Y. Liu, U(1)-decoupling, KK and BCJ relations in \( \mathcal{N}=4 \) SYM, Phys. Rev. D 82 (2010) 065001 [arXiv:1005.1821] [INSPIRE].ADSGoogle Scholar
  8. [8]
    Y.-X. Chen, Y.-J. Du and B. Feng, A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory, JHEP 02 (2011) 112 [arXiv:1101.0009] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    H. Kawai, D. Lewellen and S. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    Z. Bern, L.J. Dixon, D. Dunbar, M. Perelstein and J. 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].ADSCrossRefGoogle Scholar
  11. [11]
    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].ADSGoogle Scholar
  12. [12]
    Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N = 8 Supergravity, Phys. Rev. D 78 (2008) 105019 [arXiv:0808.4112] [INSPIRE].ADSGoogle Scholar
  13. [13]
    Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The Ultraviolet Behavior of N = 8 Supergravity at Four Loops, Phys. Rev. Lett. 103 (2009) 081301 [arXiv:0905.2326] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    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].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    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].ADSGoogle Scholar
  16. [16]
    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].ADSGoogle Scholar
  17. [17]
    C. Boucher-Veronneau and L. Dixon, N ≥ 4 Supergravity Amplitudes from Gauge Theory at Two Loops, JHEP 12 (2011) 046 [arXiv:1110.1132] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    Z. Bern, J. Carrasco, L. 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].ADSGoogle Scholar
  19. [19]
    Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Absence of Three-Loop Four-Point Divergences in N = 4 Supergravity, Phys. Rev. Lett. 108 (2012) 201301 [arXiv:1202.3423] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure, Phys. Rev. D 86 (2012) 105014 [arXiv:1209.2472] [INSPIRE].ADSGoogle Scholar
  21. [21]
    S. Oxburgh and C. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    E.Y. Yuan, Virtual Color-Kinematics Duality: 6-pt 1-Loop MHV Amplitudes, arXiv:1210.1816 [INSPIRE].
  23. [23]
    R. Saotome and R. Akhoury, Relationship Between Gravity and Gauge Scattering in the High Energy Limit, JHEP 01 (2013) 123 [arXiv:1210.8111] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    R.H. Boels and R.S. Isermann, On powercounting in perturbative quantum gravity theories through color-kinematic duality, arXiv:1212.3473 [INSPIRE].
  25. [25]
    R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    N. Bjerrum-Bohr, P.H. Damgaard, R. Monteiro and D. O’Connell, Algebras for Amplitudes, JHEP 06 (2012) 061 [arXiv:1203.0944] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    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].ADSCrossRefGoogle Scholar
  28. [28]
    R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    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].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    Z. Bern, A. De Freitas and H. Wong, On the coupling of gravitons to matter, Phys. Rev. Lett. 84 (2000) 3531 [hep-th/9912033] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    Z. Bern and T. Dennen, A Color Dual Form for Gauge-Theory Amplitudes, Phys. Rev. Lett. 107 (2011) 081601 [arXiv:1103.0312] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    F.A. Berends and W. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
  34. [34]
    N. 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].ADSGoogle Scholar
  35. [35]
    F.A. Berends and W. Giele, Multiple Soft Gluon Radiation in Parton Processes, Nucl. Phys. B 313 (1989) 595 [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Center of Mathematical ScienceZhejiang UniversityHangzhouP.R China
  2. 2.Department of ElectrophysicsNational Chiao Tung UniversityHsinchuR.O.C.
  3. 3.Department of Physics and Center for Field Theory and Particle PhysicsFudan UniversityShanghaiP.R China
  4. 4.Zhejiang Institute of Modern PhysicsZhejiang UniversityHangzhouP.R China

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