Monodromy-like relations for finite loop amplitudes

  • N. E. J. Bjerrum-Bohr
  • P. H. Damgaard
  • H. Johansson
  • T. Søndergaard
Open Access


We investigate the existence of relations for finite one-loop amplitudes in Yang-Mills theory. Using a diagrammatic formalism and a remarkable connection between tree and loop level, we deduce sequences of amplitude relations for any number of external legs.


NLO Computations Superstrings and Heterotic Strings QCD 


  1. [1]
    F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [SPIRES].ADSCrossRefGoogle Scholar
  2. [2]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev. D 75 (2007) 125019 [arXiv:0704.1835] [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    C.F. Berger et al., An automated implementation of on-shell methods for one-loop amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [SPIRES].ADSGoogle Scholar
  8. [8]
    S. Badger, B. Biedermann and P. Uwer, NGluon: a package to calculate one-loop multi-gluon amplitudes, arXiv:1011.2900 [SPIRES].
  9. [9]
    J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    A. Brandhuber, P. Heslop and G. Travaglini, A note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [SPIRES].MathSciNetADSGoogle Scholar
  12. [12]
    Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [SPIRES].MathSciNetADSGoogle Scholar
  13. [13]
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    Z. Bern and T. Dennen, A color dual form for gauge-theory amplitudes, arXiv:1103.0312 [SPIRES].
  15. [15]
    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] [SPIRES].ADSGoogle Scholar
  16. [16]
    Z. Bern, G. Chalmers, L.J. Dixon and D.A. Kosower, One loop N gluon amplitudes with maximal helicity violation via collinear limits, Phys. Rev. Lett. 72 (1994) 2134 [hep-ph/9312333] [SPIRES].ADSCrossRefGoogle Scholar
  17. [17]
    G. Mahlon, Multi-gluon helicity amplitudes involving a quark loop, Phys. Rev. D 49 (1994) 4438 [hep-ph/9312276] [SPIRES].ADSGoogle Scholar
  18. [18]
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [SPIRES].ADSCrossRefGoogle Scholar
  20. [20]
    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] [SPIRES].ADSCrossRefGoogle Scholar
  21. [21]
    S. Stieberger and T.R. Taylor, Multi-gluon scattering in open superstring theory, Phys. Rev. D 74 (2006) 126007 [hep-th/0609175] [SPIRES].MathSciNetADSGoogle Scholar
  22. [22]
    R.M. Schabinger, One-loop N = 4 super Yang-Mills scattering amplitudes to all orders in the dimensional regularization parameter, arXiv:1103.2769 [SPIRES].
  23. [23]
    S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [SPIRES].
  24. [24]
    C. Cheung, D. O’Connell and B. Wecht, BCFW recursion relations and string theory, JHEP 09 (2010) 052 [arXiv:1002.4674] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    R.H. Boels, D. Marmiroli and N.A. Obers, On-shell recursion in string theory, JHEP 10 (2010) 034 [arXiv:1002.5029] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, A recursive formula for N-point SYM tree amplitudes, arXiv:1012.3981 [SPIRES].
  27. [27]
    C.R. Mafra, Simplifying the tree-level superstring massless five-point amplitude, JHEP 01 (2010) 007 [arXiv:0909.5206] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    S.H. Henry Tye and Y. Zhang, Dual identities inside the gluon and the graviton scattering amplitudes, JHEP 06 (2010) 071 [arXiv:1003.1732] [SPIRES].ADSCrossRefGoogle Scholar
  29. [29]
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    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] [SPIRES].MathSciNetADSGoogle Scholar
  31. [31]
    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] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    T. Sondergaard, New relations for gauge-theory amplitudes with matter, Nucl. Phys. B 821 (2009) 417 [arXiv:0903.5453] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    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] [SPIRES].ADSGoogle 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] [SPIRES].ADSCrossRefGoogle Scholar
  35. [35]
    Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [SPIRES].
  37. [37]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One-loop self-dual and N = 4 super Yang-Mills, Phys. Lett. B 394 (1997) 105 [hep-th/9611127] [SPIRES].MathSciNetADSGoogle Scholar
  38. [38]
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Supergravity and the S matrix, Phys. Rev. D 15 (1977) 996 [SPIRES].ADSGoogle Scholar
  40. [40]
    Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    Z. Bern, L.J. Dixon and D.A. Kosower, New QCD results from string theory, hep-th/9311026 [SPIRES].
  42. [42]
    Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70 (1993) 2677 [hep-ph/9302280] [SPIRES].ADSCrossRefGoogle Scholar
  43. [43]
    Z. Bern, L.J. Dixon and D.A. Kosower, The last of the finite loop amplitudes in QCD, Phys. Rev. D 72 (2005) 125003 [hep-ph/0505055] [SPIRES].MathSciNetADSGoogle Scholar
  44. [44]
    Z. Bern, L.J. Dixon and D.A. Kosower, On-shell recurrence relations for one-loop QCD amplitudes, Phys. Rev. D 71 (2005) 105013 [hep-th/0501240] [SPIRES].MathSciNetADSGoogle Scholar
  45. [45]
    C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, Bootstrapping one-loop QCD amplitudes with general helicities, Phys. Rev. D 74 (2006) 036009 [hep-ph/0604195] [SPIRES].ADSGoogle Scholar

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© The Author(s) 2011

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • N. E. J. Bjerrum-Bohr
    • 1
  • P. H. Damgaard
    • 1
  • H. Johansson
    • 2
  • T. Søndergaard
    • 1
  1. 1.Niels Bohr International Academy and Discovery CenterThe Niels Bohr InstituteCopenhagenDenmark
  2. 2.Institut de Physique Théorique, CEA-SaclayGif-sur-YvetteFrance

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