Journal of High Energy Physics

, 2015:217 | Cite as

The CHY representation of tree-level primitive QCD amplitudes

  • Leonardo de la Cruz
  • Alexander Kniss
  • Stefan WeinzierlEmail author
Open Access
Regular Article - Theoretical Physics


In this paper we construct a CHY representation for all tree-level primitive QCD amplitudes. The quarks may be massless or massive. We define a generalised cyclic factor Ĉ(w, z) and a generalised permutation invariant function Ê(z, p, ε). The amplitude is then given as a contour integral encircling the solutions of the scattering equations with the product ĈÊ as integrand. Equivalently, it is given as a sum over the inequivalent solutions of the scattering equations, where the summand consists of a Jacobian times the product Ĉ Ê. This representation separates information: The generalised cyclic factor does not depend on the helicities of the external particles, the generalised permutation invariant function does not depend on the ordering of the external particles.


Scattering Amplitudes QCD 


Open Access

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  1. [1]
    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
  2. [2]
    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
  3. [3]
    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
  4. [4]
    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
  5. [5]
    F. Cachazo, S. He and E.Y. Yuan, Einstein- Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    F. Cachazo, S. He and E.Y. Yuan, New Double Soft Emission Theorems, Phys. Rev. D 92 (2015) 065030 [arXiv:1503.04816] [INSPIRE].ADSGoogle Scholar
  8. [8]
    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].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    L. Dolan and P. Goddard, The Polynomial Form of the Scattering Equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    Y.-H. He, C. Matti and C. Sun, The Scattering Variety, JHEP 10 (2014) 135 [arXiv:1403.6833] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    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
  12. [12]
    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
  13. [13]
    S.G. Naculich, Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation, JHEP 09 (2015) 122 [arXiv:1506.06134] [INSPIRE].CrossRefADSGoogle Scholar
  14. [14]
    S. Weinzierl, Fermions and the scattering equations, JHEP 03 (2015) 141 [arXiv:1412.5993] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  15. [15]
    C. Kalousios, Massless scattering at special kinematics as Jacobi polynomials, J. Phys. A 47 (2014) 215402 [arXiv:1312.7743] [INSPIRE].ADSMathSciNetGoogle Scholar
  16. [16]
    S. Weinzierl, On the solutions of the scattering equations, JHEP 04 (2014) 092 [arXiv:1402.2516] [INSPIRE].CrossRefADSGoogle Scholar
  17. [17]
    C.S. Lam, Permutation Symmetry of the Scattering Equations, Phys. Rev. D 91 (2015) 045019 [arXiv:1410.8184] [INSPIRE].ADSGoogle Scholar
  18. [18]
    R. Monteiro and D. O’Connell, The Kinematic Algebras from the Scattering Equations, JHEP 03 (2014) 110 [arXiv:1311.1151] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    F. Cachazo and H. Gomez, Computation of Contour Integrals on0,n, arXiv:1505.03571 [INSPIRE].
  20. [20]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Integration Rules for Scattering Equations, JHEP 09 (2015) 129 [arXiv:1506.06137] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Integration Rules for Loop Scattering Equations, arXiv:1508.03627 [INSPIRE].
  22. [22]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Scattering Equations and Feynman Diagrams, JHEP 09 (2015) 136 [arXiv:1507.00997] [INSPIRE].CrossRefADSGoogle Scholar
  23. [23]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, P. Tourkine and P. Vanhove, Scattering Equations and String Theory Amplitudes, Phys. Rev. D 90 (2014) 106002 [arXiv:1403.4553] [INSPIRE].ADSGoogle Scholar
  24. [24]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].CrossRefADSGoogle Scholar
  25. [25]
    N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    H. Gomez and E.Y. Yuan, N-point tree-level scattering amplitude in the new Berkovits‘ string, JHEP 04 (2014) 046 [arXiv:1312.5485] [INSPIRE].CrossRefADSGoogle Scholar
  27. [27]
    T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor Strings in Four Dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].CrossRefADSGoogle Scholar
  29. [29]
    E. Casali and P. Tourkine, Infrared behaviour of the one-loop scattering equations and supergravity integrands, JHEP 04 (2015) 013 [arXiv:1412.3787] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  30. [30]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].CrossRefADSGoogle Scholar
  31. [31]
    B.U.W. Schwab and A. Volovich, Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations, Phys. Rev. Lett. 113 (2014) 101601 [arXiv:1404.7749] [INSPIRE].CrossRefADSGoogle Scholar
  32. [32]
    N. Afkhami-Jeddi, Soft Graviton Theorem in Arbitrary Dimensions, arXiv:1405.3533 [INSPIRE].
  33. [33]
    M. Zlotnikov, Sub-sub-leading soft-graviton theorem in arbitrary dimension, JHEP 10 (2014) 148 [arXiv:1407.5936] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  34. [34]
    C. Kalousios and F. Rojas, Next to subleading soft-graviton theorem in arbitrary dimensions, JHEP 01 (2015) 107 [arXiv:1407.5982] [INSPIRE].CrossRefADSGoogle Scholar
  35. [35]
    C.D. White, Diagrammatic insights into next-to-soft corrections, Phys. Lett. B 737 (2014) 216 [arXiv:1406.7184] [INSPIRE].CrossRefADSGoogle Scholar
  36. [36]
    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
  37. [37]
    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
  38. [38]
    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
  39. [39]
    H. Johansson and A. Ochirov, Color-Kinematics Duality for QCD Amplitudes, arXiv:1507.00332 [INSPIRE].
  40. [40]
    L. de la Cruz, A. Kniss and S. Weinzierl, Proof of the fundamental BCJ relations for QCD amplitudes, JHEP 09 (2015) 197 [arXiv:1508.01432] [INSPIRE].CrossRefADSGoogle Scholar
  41. [41]
    F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].CrossRefADSGoogle Scholar
  42. [42]
    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
  43. [43]
    Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to two quark three gluon amplitudes, Nucl. Phys. B 437 (1995) 259 [hep-ph/9409393] [INSPIRE].
  44. [44]
    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
  45. [45]
    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
  46. [46]
    S. Stieberger, Open and Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
  47. [47]
    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
  48. [48]
    T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88 (2013) 014020 [arXiv:1304.7809] [INSPIRE].ADSGoogle Scholar
  49. [49]
    T. Melia, Getting more flavor out of one-flavor QCD, Phys. Rev. D 89 (2014) 074012 [arXiv:1312.0599] [INSPIRE].ADSGoogle Scholar
  50. [50]
    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
  51. [51]
    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].ADSGoogle Scholar
  52. [52]
    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].CrossRefADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Leonardo de la Cruz
    • 1
  • Alexander Kniss
    • 1
  • Stefan Weinzierl
    • 1
    Email author
  1. 1.PRISMA Cluster of Excellence, Institut für PhysikJohannes Gutenberg-Universität MainzMainzGermany

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