From six to four and more: massless and massive maximal super Yang-Mills amplitudes in 6d and 4d and their hidden symmetries

  • Jan PlefkaEmail author
  • Theodor Schuster
  • Valentin Verschinin
Open Access
Regular Article - Theoretical Physics


A self-consistent exposition of the theory of tree-level superamplitudes of the 4d \( \mathcal{N}=4 \) and 6d \( \mathcal{N}=\left(1,1\right) \) maximally supersymmetric Yang-Mills theories is provided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the \( \mathcal{N}=4 \) SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the standard and hidden symmetries of the tree-level superamplitudes of \( \mathcal{N}=\left(1,1\right) \) SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in \( \mathcal{N}=\left(1,1\right) \) SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five- and six-point superamplitudes as well as arbitrary multiplicity formulae valid for certain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the \( \mathcal{N}=4 \) SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional \( \mathcal{N}=\left(1,1\right) \) SYM theory. We exploit this correspondence to construct the super-Poincaré and enhanced dual conformal symmetries of massive tree superamplitudes in \( \mathcal{N}=4 \) SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d massless superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five.


Supersymmetric gauge theory Scattering Amplitudes Extended Supersymmetry Space-Time Symmetries 


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.


  1. [1]
    A.E. Lipstein and L. Mason, Amplitudes of 3d Yang-Mills Theory, JHEP 01 (2013) 009 [arXiv:1207.6176] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    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] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    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] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    T. Dennen and Y.-t. Huang, Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes, JHEP 01 (2011) 140 [arXiv:1010.5874] [INSPIRE].
  5. [5]
    S. Caron-Huot and D. O’Connell, Spinor Helicity and Dual Conformal Symmetry in Ten Dimensions, JHEP 08 (2011) 014 [arXiv:1010.5487] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in \( \mathcal{N}=4 \) super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    J.M. Drummond and J.M. Henn, All tree-level amplitudes in \( \mathcal{N}=4 \) SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    N. Craig, H. Elvang, M. Kiermaier and T. Slatyer, Massive amplitudes on the Coulomb branch of \( \mathcal{N}=4 \) SYM, JHEP 12 (2011) 097 [arXiv:1104.2050] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of \( \mathcal{N}=4 \) super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    Y.-t. Huang, Non-Chiral S-matrix of \( \mathcal{N}=4 \) Super Yang-Mills, arXiv:1104.2021 [INSPIRE].
  12. [12]
    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].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    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].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    M. Bianchi, H. Elvang and D.Z. Freedman, Generating Tree Amplitudes in \( \mathcal{N}=4 \) SYM and \( \mathcal{N}=8 \) SG, JHEP 09 (2008) 063 [arXiv:0805.0757] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  17. [17]
    T. Dennen, Y.-t. Huang and W. Siegel, Supertwistor space for 6D maximal super Yang-Mills, JHEP 04 (2010) 127 [arXiv:0910.2688] [INSPIRE].
  18. [18]
    Z. Bern, J.J. Carrasco, T. Dennen, Y.-t. Huang and H. Ita, Generalized Unitarity and Six-Dimensional Helicity, Phys. Rev. D 83 (2011) 085022 [arXiv:1010.0494] [INSPIRE].
  19. [19]
    A. Brandhuber, D. Korres, D. Koschade and G. Travaglini, One-loop Amplitudes in Six-Dimensional (1, 1) Theories from Generalised Unitarity, JHEP 02 (2011) 077 [arXiv:1010.1515] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
  21. [21]
    J.M. Henn and J.C. Plefka, Lecture Notes in Physics. Vol. 883: Scattering Amplitudes in Gauge Theories, Springer, Heidelberg Germany (2014).Google Scholar
  22. [22]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
  23. [23]
    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
  24. [24]
    T. Schuster, Color ordering in QCD, Phys. Rev. D 89 (2014) 105022 [arXiv:1311.6296] [INSPIRE].ADSGoogle Scholar
  25. [25]
    P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Multiple Bremsstrahlung in Gauge Theories at High-Energies. 1. General Formalism for Quantum Electrodynamics, Nucl. Phys. B 206 (1982) 53 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    F.A. Berends et al., Multiple Bremsstrahlung in Gauge Theories at High-Energies. 2. Single Bremsstrahlung, Nucl. Phys. B 206 (1982) 61 [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    R. Kleiss and W.J. Stirling, Spinor Techniques for Calculating \( p\overline{p}\to {W}_{\pm }/{Z}_0 \) + Jets, Nucl. Phys. B 262 (1985) 235 [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    Z. Xu, D.-H. Zhang and L. Chang, Helicity Amplitudes for Multiple Bremsstrahlung in Massless Nonabelian Gauge Theories, Nucl. Phys. B 291 (1987) 392 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    C. Cheung and D. O’Connell, Amplitudes and Spinor-Helicity in Six Dimensions, JHEP 07 (2009) 075 [arXiv:0902.0981] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    G. Georgiou, E.W.N. Glover and V.V. Khoze, Non-MHV tree amplitudes in gauge theory, JHEP 07 (2004) 048 [hep-th/0407027] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    N. Arkani-Hamed, F. Cachazo and C. Cheung, The Grassmannian Origin Of Dual Superconformal Invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    N. Arkani-Hamed et al., Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 [INSPIRE].
  35. [35]
    N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 1410 (2014) 30 [arXiv:1312.2007] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    Y.-t. Huang and A.E. Lipstein, Amplitudes of 3D and 6D Maximal Superconformal Theories in Supertwistor Space, JHEP 10 (2010) 007 [arXiv:1004.4735] [INSPIRE].
  37. [37]
    H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in \( \mathcal{N}<4 \) SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].
  38. [38]
    Z. Bern, L.J. Dixon and D.A. Kosower, On-Shell Methods in Perturbative QCD, Annals Phys. 322 (2007) 1587 [arXiv:0704.2798] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    C. Cheung, On-Shell Recursion Relations for Generic Theories, JHEP 03 (2010) 098 [arXiv:0808.0504] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    N. Arkani-Hamed and J. Kaplan, On Tree Amplitudes in Gauge Theory and Gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  41. [41]
    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].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    T. Melia, Getting more flavour out of one-flavour QCD, Phys. Rev. D 89 (2014) 074012 [arXiv:1312.0599] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Jan Plefka
    • 1
    • 2
    Email author
  • Theodor Schuster
    • 1
  • Valentin Verschinin
    • 1
    • 3
  1. 1.Institut für Physik and IRIS AdlershofHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Institut für Theoretische PhysikEidgenössische Technische Hochschule ZürichZürichSwitzerland
  3. 3.Centre de Physique Théorique, École Polytechnique, CNRSPalaiseauFrance

Personalised recommendations