Journal of High Energy Physics

, 2016:170 | Cite as

Color-kinematics duality for QCD amplitudes

  • Henrik Johansson
  • Alexander Ochirov
Open Access
Regular Article - Theoretical Physics


We show that color-kinematics duality is present in tree-level amplitudes of quantum chromodynamics with massive flavored quarks. Starting with the color structure of QCD, we work out a new color decomposition for n-point tree amplitudes in a reduced basis of primitive amplitudes. These primitives, with k quark-antiquark pairs and (n − 2k) gluons, are taken in the (n − 2)!/k! Melia basis, and are independent under the color-algebra Kleiss-Kuijf relations. This generalizes the color decomposition of Del Duca, Dixon, and Maltoni to an arbitrary number of quarks. The color coefficients in the new decomposition are given by compact expressions valid for arbitrary gauge group and representation. Considering the kinematic structure, we show through explicit calculations that color-kinematics duality holds for amplitudes with general configurations of gluons and massive quarks. The new (massive) amplitude relations that follow from the duality can be mapped to a well-defined subset of the familiar BCJ relations for gluons. They restrict the amplitude basis further down to (n − 3)!(2k − 2)/k! primitives, for two or more quark lines. We give a decomposition of the full amplitude in that basis. The presented results provide strong evidence that QCD obeys the color-kinematics duality, at least at tree level. The results are also applicable to supersymmetric and D-dimensional extensions of QCD.


Scattering Amplitudes QCD Gauge Symmetry 


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.

Supplementary material

13130_2016_3032_MOESM1_ESM.nb (58 kb)
ESM 1 (NB 58 kb)
13130_2016_3032_MOESM2_ESM.nb (62 kb)
ESM 2 (NB 61 kb)


  1. [1]
    Z. Bern, L.J. Dixon, F. Febres Cordero, S. Höche, H. Ita, D.A. Kosower et al., Next-to-Leading Order W + 5-Jet Production at the LHC, Phys. Rev. D 88 (2013) 014025 [arXiv:1304.1253] [INSPIRE].ADSGoogle Scholar
  2. [2]
    C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].CrossRefADSGoogle Scholar
  3. [3]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].CrossRefADSGoogle Scholar
  5. [5]
    Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280] [INSPIRE].CrossRefADSGoogle Scholar
  6. [6]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    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].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    R.K. Ellis, W.T. Giele and Z. Kunszt, A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes, JHEP 03 (2008) 003 [arXiv:0708.2398] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde, H. Ita et al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].ADSGoogle Scholar
  10. [10]
    G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: A Program implementing the OPP reduction method to compute one-loop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596] [INSPIRE].CrossRefADSGoogle Scholar
  11. [11]
    P. Mastrolia, G. Ossola, C.G. Papadopoulos and R. Pittau, Optimizing the Reduction of One-Loop Amplitudes, JHEP 06 (2008) 030 [arXiv:0803.3964] [INSPIRE].CrossRefADSGoogle Scholar
  12. [12]
    W.T. Giele and G. Zanderighi, On the Numerical Evaluation of One-Loop Amplitudes: The Gluonic Case, JHEP 06 (2008) 038 [arXiv:0805.2152] [INSPIRE].CrossRefADSGoogle Scholar
  13. [13]
    R.K. Ellis, W.T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop amplitudes for W + 3 jet production in hadron collisions, JHEP 01 (2009) 012 [arXiv:0810.2762] [INSPIRE].CrossRefADSGoogle Scholar
  14. [14]
    C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg et al., Precise Predictions for W + 3 Jet Production at Hadron Colliders, Phys. Rev. Lett. 102 (2009) 222001 [arXiv:0902.2760] [INSPIRE].CrossRefADSGoogle Scholar
  15. [15]
    G. Bevilacqua, M. Czakon, C.G. Papadopoulos, R. Pittau and M. Worek, Assault on the NLO Wishlist: \( pp\to t\overline{t}b\overline{b} \), JHEP 09 (2009) 109 [arXiv:0907.4723] [INSPIRE].CrossRefADSGoogle Scholar
  16. [16]
    P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, Scattering AMplitudes from Unitarity-based Reduction Algorithm at the Integrand-level, JHEP 08 (2010) 080 [arXiv:1006.0710] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  17. [17]
    C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg et al., Precise Predictions for W + 4 Jet Production at the Large Hadron Collider, Phys. Rev. Lett. 106 (2011) 092001 [arXiv:1009.2338] [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    S. Badger, B. Biedermann and P. Uwer, NGluon: A Package to Calculate One-loop Multi-gluon Amplitudes, Comput. Phys. Commun. 182 (2011) 1674 [arXiv:1011.2900] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  19. [19]
    V. Hirschi, R. Frederix, S. Frixione, M.V. Garzelli, F. Maltoni and R. Pittau, Automation of one-loop QCD corrections, JHEP 05 (2011) 044 [arXiv:1103.0621] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  20. [20]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].CrossRefADSGoogle Scholar
  22. [22]
    C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD, Phys. Rev. Lett. 91 (2003) 182002 [hep-ph/0306192] [INSPIRE].CrossRefADSGoogle Scholar
  23. [23]
    V. A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1.CrossRefMathSciNetzbMATHGoogle Scholar
  24. [24]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  25. [25]
    C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].CrossRefADSGoogle Scholar
  26. [26]
    F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].CrossRefADSGoogle Scholar
  27. [27]
    F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    L.J. Dixon, J.M. Henn, J. Plefka and T. Schuster, All tree-level amplitudes in massless QCD, JHEP 01 (2011) 035 [arXiv:1010.3991] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  30. [30]
    R. Kleiss and H. Kuijf, Multi - Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].CrossRefADSGoogle Scholar
  31. [31]
    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].ADSMathSciNetGoogle Scholar
  32. [32]
    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].CrossRefADSMathSciNetGoogle Scholar
  33. [33]
    V. Del Duca, A. Frizzo and F. Maltoni, Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit, Nucl. Phys. B 568 (2000) 211 [hep-ph/9909464] [INSPIRE].CrossRefADSGoogle Scholar
  34. [34]
    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].CrossRefADSGoogle Scholar
  35. [35]
    D. Kosower, B.-H. Lee and V.P. Nair, Multi gluon scattering: a string based calculation, Phys. Lett. B 201 (1988) 85 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    M.L. Mangano, The Color Structure of Gluon Emission, Nucl. Phys. B 309 (1988) 461 [INSPIRE].CrossRefADSGoogle Scholar
  37. [37]
    T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88 (2013) 014020 [arXiv:1304.7809] [INSPIRE].ADSGoogle Scholar
  38. [38]
    T. Melia, Getting more flavor out of one-flavor QCD, Phys. Rev. D 89 (2014) 074012 [arXiv:1312.0599] [INSPIRE].ADSGoogle Scholar
  39. [39]
    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
  40. [40]
    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
  41. [41]
    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
  42. [42]
    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].ADSGoogle Scholar
  43. [43]
    J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled \( \mathcal{N}\le 4 \) supergravity, JHEP 03 (2013) 056 [arXiv:1212.1146] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  44. [44]
    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].CrossRefADSMathSciNetGoogle Scholar
  45. [45]
    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].CrossRefADSMathSciNetGoogle Scholar
  46. [46]
    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, Phys. Rev. D 92 (2015) 045041 [arXiv:1303.6605] [INSPIRE].ADSGoogle Scholar
  47. [47]
    J. Nohle, Color-Kinematics Duality in One-Loop Four-Gluon Amplitudes with Matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].ADSGoogle Scholar
  48. [48]
    Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].CrossRefADSGoogle Scholar
  49. [49]
    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].CrossRefADSMathSciNetGoogle Scholar
  50. [50]
    H. Johansson and A. Ochirov, Pure Gravities via Color-Kinematics Duality for Fundamental Matter, JHEP 11 (2015) 046 [arXiv:1407.4772] [INSPIRE].CrossRefADSGoogle Scholar
  51. [51]
    Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in \( \mathcal{N}=5 \) supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].ADSGoogle Scholar
  52. [52]
    T. Sondergaard, New Relations for Gauge-Theory Amplitudes with Matter, Nucl. Phys. B 821 (2009) 417 [arXiv:0903.5453] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  53. [53]
    S. Weinzierl, Fermions and the scattering equations, JHEP 03 (2015) 141 [arXiv:1412.5993] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  54. [54]
    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].CrossRefADSGoogle Scholar
  55. [55]
    S.G. Naculich, CHY representations for gauge theory and gravity amplitudes with up to three massive particles, JHEP 05 (2015) 050 [arXiv:1501.03500] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  56. [56]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
  57. [57]
    F.A. Berends and W. Giele, The Six Gluon Process as an Example of Weyl-Van Der Waerden Spinor Calculus, Nucl. Phys. B 294 (1987) 700 [INSPIRE].CrossRefADSGoogle Scholar
  58. [58]
    M.L. Mangano, S.J. Parke and Z. Xu, Duality and Multi - Gluon Scattering, Nucl. Phys. B 298 (1988) 653 [INSPIRE].CrossRefADSGoogle Scholar
  59. [59]
    Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  60. [60]
    P. Cvitanovic, P.G. Lauwers and P.N. Scharbach, Gauge Invariance Structure of Quantum Chromodynamics, Nucl. Phys. B 186 (1981) 165 [INSPIRE].CrossRefADSGoogle Scholar
  61. [61]
    D.A. Kosower, Color Factorization for Fermionic Amplitudes, Nucl. Phys. B 315 (1989) 391 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  62. [62]
    M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].CrossRefADSGoogle Scholar
  63. [63]
    R.K. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts, Phys. Rept. 518 (2012) 141 [arXiv:1105.4319] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  64. [64]
    H. Ita and K. Ozeren, Colour Decompositions of Multi-quark One-loop QCD Amplitudes, JHEP 02 (2012) 118 [arXiv:1111.4193] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  65. [65]
    S. Badger, B. Biedermann, P. Uwer and V. Yundin, Numerical evaluation of virtual corrections to multi-jet production in massless QCD, Comput. Phys. Commun. 184 (2013) 1981 [arXiv:1209.0100] [INSPIRE].CrossRefADSGoogle Scholar
  66. [66]
    T. Schuster, Color ordering in QCD, Phys. Rev. D 89 (2014) 105022 [arXiv:1311.6296] [INSPIRE].ADSGoogle Scholar
  67. [67]
    C. Reuschle and S. Weinzierl, Decomposition of one-loop QCD amplitudes into primitive amplitudes based on shuffle relations, Phys. Rev. D 88 (2013) 105020 [arXiv:1310.0413] [INSPIRE].ADSGoogle Scholar
  68. [68]
    R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  69. [69]
    M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Scattering amplitudes in \( \mathcal{N}=2 \) Maxwell-Einstein and Yang-Mills/Einstein supergravity, JHEP 01 (2015) 081 [arXiv:1408.0764] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  70. [70]
    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].CrossRefADSMathSciNetGoogle Scholar
  71. [71]
    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].CrossRefADSMathSciNetGoogle Scholar
  72. [72]
    S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
  73. [73]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  74. [74]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  75. [75]
    C.R. Mafra and O. Schlotterer, The Structure of n-Point One-Loop Open Superstring Amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].CrossRefADSGoogle Scholar
  76. [76]
    J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  77. [77]
    S. Stieberger and T.R. Taylor, Closed String Amplitudes as Single-Valued Open String Amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  78. [78]
    C.R. Mafra and O. Schlotterer, Multiparticle SYM equations of motion and pure spinor BRST blocks, JHEP 07 (2014) 153 [arXiv:1404.4986] [INSPIRE].CrossRefADSGoogle Scholar
  79. [79]
    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].
  80. [80]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  81. [81]
    L.A. Barreiro and R. Medina, RNS derivation of N-point disk amplitudes from the revisited S-matrix approach, Nucl. Phys. B 886 (2014) 870 [arXiv:1310.5942] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  82. [82]
    A. Ochirov and P. Tourkine, BCJ duality and double copy in the closed string sector, JHEP 05 (2014) 136 [arXiv:1312.1326] [INSPIRE].CrossRefADSGoogle Scholar
  83. [83]
    C.R. Mafra and O. Schlotterer, Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, JHEP 10 (2015) 124 [arXiv:1505.02746] [INSPIRE].CrossRefADSGoogle Scholar
  84. [84]
    R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  85. [85]
    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].CrossRefADSMathSciNetGoogle Scholar
  86. [86]
    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
  87. [87]
    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
  88. [88]
    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
  89. [89]
    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
  90. [90]
    T. Melia, Dyck words and multi-quark amplitudes, PoS(RADCOR 2013)031.
  91. [91]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
  92. [92]
    M. Kiermaier, Gravity as the Square of Gauge Theory talk at Amplitudes 2010, 4–7 May 2010˜theory/Amplitudes2010/.
  93. [93]
    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].CrossRefADSMathSciNetzbMATHGoogle Scholar
  94. [94]
    S. Badger, G. Mogull, A. Ochirov and D. O’Connell, A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory, JHEP 10 (2015) 064 [arXiv:1507.08797] [INSPIRE].CrossRefADSGoogle Scholar
  95. [95]
    Y.-J. Du, M. Sjödahl and J. Thorén, Recursion in multiplet bases for tree-level MHV gluon amplitudes, JHEP 05 (2015) 119 [arXiv:1503.00530] [INSPIRE].CrossRefADSGoogle Scholar
  96. [96]
    L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  97. [97]
    E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [arXiv:1008.0098] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  98. [98]
    E. Gardi, J.M. Smillie and C.D. White, On the renormalization of multiparton webs, JHEP 09 (2011) 114 [arXiv:1108.1357] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  99. [99]
    E. Gardi, J.M. Smillie and C.D. White, The Non-Abelian Exponentiation theorem for multiple Wilson lines, JHEP 06 (2013) 088 [arXiv:1304.7040] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  100. [100]
    M. Dukes, E. Gardi, H. McAslan, D.J. Scott and C.D. White, Webs and Posets, JHEP 01 (2014) 024 [arXiv:1310.3127] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Theory Division, Physics Department, CERNGeneva 23Switzerland
  2. 2.Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  3. 3.Nordita, KTH Royal Institute of Technology and Stockholm UniversityStockholmSweden
  4. 4.Higgs Centre for Theoretical Physics, School of Physics and AstronomyThe University of EdinburghEdinburghU.K.

Personalised recommendations