Scattering amplitudes and BCFW recursion in twistor space

  • Lionel Mason
  • David Skinner
Open Access


Twistor ideas have led to a number of recent advances in our understanding of scattering amplitudes. Much of this work has been indirect, determining the twistor space support of scattering amplitudes by examining the amplitudes in momentum space. In this paper, we construct the actual twistor scattering amplitudes themselves. We show that the recursion relations of Britto, Cachazo, Feng and Witten have a natural twistor formulation that, together with the three-point seed amplitudes, allows us to recursively construct general tree amplitudes in twistor space. We obtain explicit formulae for n-particle MHV and NMHV super-amplitudes, their CPT conjugates (whose representations are distinct in our chiral framework), and the eight particle N2MHV super-amplitude. We also give simple closed form formulae for the \( \mathcal{N} = 8 \) supergravity recursion and the MHV and \( \overline {\text{MHV}} \) amplitudes. This gives a formulation of scattering amplitudes in maximally supersymmetric theories in which superconformal symmetry and its breaking is manifest.

For N k MHV, the amplitudes are given by 2n − 4 integrals in the form of Hilbert transforms of a product of nk − 2 purely geometric, superconformally invariant twistor delta functions, dressed by certain sign operators. These sign operators subtly violate conformal invariance, even for tree-level amplitudes in \( \mathcal{N} = 4 \) super Yang-Mills, and we trace their origin to a topological property of split signature space-time. We develop the twistor transform to relate our work to the ambidextrous twistor diagram approach of Hodges and of Arkani-Hamed, Cachazo, Cheung and Kaplan.


Supersymmetric gauge theory Duality in Gauge Field Theories 


  1. [1]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. [2]
    R. Roiban, M. Spradlin and A. Volovich, On the tree-level S-matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [SPIRES].MathSciNetADSGoogle Scholar
  3. [3]
    F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [SPIRES].CrossRefADSGoogle Scholar
  4. [4]
    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].CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    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].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    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].CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [SPIRES].MathSciNetADSGoogle Scholar
  8. [8]
    Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [SPIRES].MathSciNetADSGoogle Scholar
  10. [10]
    E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    F. Cachazo and D. Skinner, On the structure of scattering amplitudes in N = 4 super Yang-Mills and N = 8 supergravity, arXiv:0801.4574 [SPIRES].
  12. [12]
    F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988 [SPIRES].
  13. [13]
    F. Cachazo, M. Spradlin and A. Volovich, Leading singularities of the two-loop six-particle MHV amplitude, Phys. Rev. D 78 (2008) 105022 [arXiv:0805.4832] [SPIRES].ADSGoogle Scholar
  14. [14]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Generalized unitarity for N = 4 super-amplitudes, arXiv:0808.0491 [SPIRES].
  15. [15]
    F. Cachazo, P. Svrček and E. Witten, Twistor space structure of one-loop amplitudes in gauge theory, JHEP 10 (2004) 074 [hep-th/0406177] [SPIRES].CrossRefADSGoogle Scholar
  16. [16]
    F. Cachazo, P. Svrček and E. Witten, Gauge theory amplitudes in twistor space and holomorphic anomaly, JHEP 10 (2004) 077 [hep-th/0409245] [SPIRES].CrossRefADSGoogle Scholar
  17. [17]
    I. Bena, Z. Bern and D.A. Kosower, Twistor-space recursive formulation of gauge theory amplitudes, Phys. Rev. D 71 (2005) 045008 [hep-th/0406133] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    I. Bena, Z. Bern, D.A. Kosower and R. Roiban, Loops in twistor space, Phys. Rev. D 71 (2005) 106010 [hep-th/0410054] [SPIRES].MathSciNetADSGoogle Scholar
  19. [19]
    J. Bedford, A. Brandhuber, B.J. Spence and G. Travaglini, A twistor approach to one-loop amplitudes in N = 1 supersymmetric Yang-Mills theory, Nucl. Phys. B 706 (2005) 100 [hep-th/0410280] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    R. Britto, F. Cachazo and B. Feng, Coplanarity in twistor space of N = 4 next-to-MHV one-loop amplitude coefficients, Phys. Lett. B 611 (2005) 167 [hep-th/0411107] [SPIRES].MathSciNetADSGoogle Scholar
  21. [21]
    S.J. Bidder, N.E.J. Bjerrum-Bohr, D.C. Dunbar and W.B. Perkins, Twistor space structure of the box coefficients of N = 1 one-loop amplitudes, Phys. Lett. B 608 (2005) 151 [hep-th/0412023] [SPIRES].MathSciNetADSGoogle Scholar
  22. [22]
    E.T. Newman, Heaven and its properties, Gen. Rel. Grav. 7 (1976) 107 [SPIRES].CrossRefADSGoogle Scholar
  23. [23]
    R. Penrose, Non-linear gravitons and curved twistor theory, Gen. Rel. Grav. 7 (1976) 31 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  24. [24]
    R.S. Ward, On self-dual gauge fields, Phys. Lett. A 61 (1977) 81 [SPIRES].ADSGoogle Scholar
  25. [25]
    L.J. Mason and N.M.J. Woodhouse, Integrability, self-duality and twistor theory, London Mathematical Society Monographs, new series 15, Oxford University Press, Oxford U.K. (1996) [SPIRES].zbMATHGoogle Scholar
  26. [26]
    L.J. Mason, Twistor actions for non-self-dual fields: a derivation of twistor-string theory, JHEP 10 (2005) 009 [hep-th/0507269] [SPIRES].CrossRefADSGoogle Scholar
  27. [27]
    R. Boels, L. Mason and D. Skinner, Supersymmetric gauge theories in twistor space, JHEP 02 (2007) 014 [hep-th/0604040] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    L.J. Mason and M. Wolf, A twistor action for N = 8 self-dual supergravity, Commun. Math. Phys. 288 (2009) 97 [arXiv:0706.1941] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. [29]
    L. Mason and D. Skinner, Gravity, twistors and the MHV formalism, arXiv:0808.3907 [SPIRES].
  30. [30]
    R. Boels, L. Mason and D. Skinner, From twistor actions to MHV diagrams, Phys. Lett. B 648 (2007) 90 [hep-th/0702035] [SPIRES].MathSciNetADSGoogle Scholar
  31. [31]
    J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [SPIRES].CrossRefADSGoogle Scholar
  32. [32]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [SPIRES].CrossRefGoogle Scholar
  33. [33]
    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].CrossRefGoogle Scholar
  34. [34]
    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
  35. [35]
    J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  36. [36]
    Z. Bern et al., The two-loop six-gluon MHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [SPIRES].MathSciNetADSGoogle Scholar
  37. [37]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  38. [38]
    R. Penrose, The central programme of twistor theory, Chaos Solitons Fractals 10 (1999) 581 [SPIRES].zbMATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    R. Penrose and M.A.H. MacCallum, Twistor theory: an approach to the quantization of fields and space-time, Phys. Rept. 6 (1972) 241 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    A.P. Hodges and S. Huggett, Twistor diagrams, Surveys High Energ. Phys. 1 (1980) 333 [SPIRES].Google Scholar
  41. [41]
    F. John, The ultra-hyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938) 300 reprinted in 75 years of the Radon transform (Vienna Austria 1992), Conf. Proc. Lecture Notes Math. Phys. 4 (1994) 301, International Press, U.S.A. (1994).CrossRefMathSciNetGoogle Scholar
  42. [42]
    M.F. Atiyah, Geometry of Yang-Mills fields, Accademia Nazionale dei Lincei Scuola Normale Superiore, Lezione Fermiane, Pisa Italy (1979).zbMATHGoogle Scholar
  43. [43]
    A.P. Hodges, Twistor diagram recursion for all gauge-theoretic tree amplitudes, hep-th/0503060 [SPIRES].
  44. [44]
    A.P. Hodges, Twistor diagrams for all tree amplitudes in gauge theory: a helicity-independent formalism, hep-th/0512336 [SPIRES].
  45. [45]
    A.P. Hodges, Scattering amplitudes for eight gauge fields, hep-th/0603101 [SPIRES].
  46. [46]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in twistor space, arXiv:0903.2110 [SPIRES].
  47. [47]
    P. Benincasa, C. Boucher-Veronneau and F. Cachazo, Taming tree amplitudes in general relativity, JHEP 11 (2007) 057 [hep-th/0702032] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  48. [48]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, arXiv:0808.1446 [SPIRES].
  49. [49]
    R. Boels, A quantization of twistor Yang-Mills theory through the background field method, Phys. Rev. D 76 (2007) 105027 [hep-th/0703080] [SPIRES].MathSciNetADSGoogle Scholar
  50. [50]
    A. Brandhuber, B.J. Spence and G. Travaglini, One-loop gauge theory amplitudes in N = 4 super Yang-Mills from MHV vertices, Nucl. Phys. B 706 (2005) 150 [hep-th/0407214] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  51. [51]
    Z. Bern, L.J. Dixon and D.A. Kosower, All next-to-maximally helicity-violating one-loop gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev. D 72 (2005) 045014 [hep-th/0412210] [SPIRES].MathSciNetADSGoogle Scholar
  52. [52]
    G.P. Korchemsky and E. Sokatchev, Twistor transform of all tree amplitudes in N = 4 SYM theory, arXiv:0907.4107 [SPIRES].
  53. [53]
    J.M. Drummond, M. Spradlin, A. Volovich and C. Wen, Tree-level amplitudes in N = 8 supergravity, Phys. Rev. D 79 (2009) 105018 [arXiv:0901.2363] [SPIRES].MathSciNetADSGoogle Scholar
  54. [54]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, arXiv:0907.5418 [SPIRES].
  55. [55]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [SPIRES].
  56. [56]
    L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [SPIRES].CrossRefGoogle Scholar
  57. [57]
    N. Arkani-Hamed, F. Cachazo and C. Cheung, The Grassmannian origin of dual superconformal invariance, arXiv:0909.0483 [SPIRES].
  58. [58]
    M. Bullimore, L. Mason and D. Skinner, Twistor-strings, Grassmannians and leading singularities, arXiv:0912.0539 [SPIRES].
  59. [59]
    J. Kaplan, Unraveling L n,k : Grassmannian kinematics, arXiv:0912.0957 [SPIRES].
  60. [60]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  61. [61]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  62. [62]
    H. Kawai, D.C. Lewellen and S.H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [SPIRES].CrossRefADSGoogle Scholar
  63. [63]
    H. Elvang and D.Z. Freedman, Note on graviton MHV amplitudes, JHEP 05 (2008) 096 [arXiv:0710.1270] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  64. [64]
    L.J. Mason and D. Skinner, An ambitwistor Yang-Mills Lagrangian, Phys. Lett. B 636 (2006) 60 [hep-th/0510262] [SPIRES].MathSciNetADSGoogle Scholar
  65. [65]
    K. Risager, A direct proof of the CSW rules, JHEP 12 (2005) 003 [hep-th/0508206] [SPIRES].CrossRefADSGoogle Scholar
  66. [66]
    H. Elvang, D.Z. Freedman and M. Kiermaier, Proof of the MHV vertex expansion for all tree amplitudes in N = 4 SYM theory, JHEP 06 (2009) 068 [arXiv:0811.3624] [SPIRES].CrossRefADSGoogle Scholar
  67. [67]
    M. Kiermaier and S.G. Naculich, A super MHV vertex expansion for N = 4 SYM theory, JHEP 05 (2009) 072 [arXiv:0903.0377] [SPIRES].CrossRefADSGoogle Scholar
  68. [68]
    V.P. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215 [SPIRES].ADSGoogle Scholar
  69. [69]
    I.M. Gel’fand and G.E. Shilov, Generalised functions, Academic Press, U.S.A. (1964) [ISBN:0122795016].Google Scholar
  70. [70]
    M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 supergravity as limits of string theories, Nucl. Phys. B 198 (1982) 474 [SPIRES].CrossRefADSGoogle Scholar
  71. [71]
    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].CrossRefMathSciNetADSGoogle Scholar
  72. [72]
    R. Penrose, The universal bracket factor, in Advances in Twistor Theory, Pitman Research Notes in Maths 37, U.S.A. (1979).Google Scholar
  73. [73]
    R. Penrose and W. Rindler, Spinors and space-time, volume I, Cambridge University Press, Cambridge U.K. (1984).zbMATHGoogle Scholar
  74. [74]
    R. Penrose and W. Rindler, Spinors and space-time, volume II, Cambridge University Press, Cambridge U.K. (1986).Google Scholar
  75. [75]
    A. Hodges, Elemental states, section 1.5.11 in Further advances in twistor theory, volume I, L.J. Mason and L.P. Hughston eds., Pitman Research Notes in Math. 231, U.S.A. (1986).Google Scholar
  76. [76]
    M.G. Eastwood, R. Penrose and R.O. Wells, Cohomology and massless fields, Commun. Math. Phys. 78 (1981) 305 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  77. [77]
    T.N. Bailey, M.G. Eastwood, R. Gover and L.J. Mason, Complex analysis and the Funk transform, J. Korean Math. Soc. 40 (2003) 577.zbMATHMathSciNetGoogle Scholar
  78. [78]
    W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Proc. Sympos. Pure Math. 62 Part 2 (1997) [alg-geom/9608011].
  79. [79]
    S. Gukov, L. Motl and A. Neitzke, Equivalence of twistor prescriptions for super Yang-Mills, Adv. Theor. Math. Phys. 11 (2007) 199 [hep-th/0404085] [SPIRES].zbMATHMathSciNetGoogle Scholar
  80. [80]
    C. Vergu, On the factorisation of the connected prescription for Yang-Mills amplitudes, Phys. Rev. D 75 (2007) 025028 [hep-th/0612250] [SPIRES].MathSciNetADSGoogle Scholar
  81. [81]
    L. Dolan, C.R. Nappi and E. Witten, Yangian symmetry in D = 4 superconformal Yang-Mills theory, hep-th/0401243 [SPIRES].
  82. [82]
    I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [SPIRES].MathSciNetADSGoogle Scholar
  83. [83]
    N. Beisert, R. Ricci, A.A. Tseytlin and M. Wolf, Dual superconformal symmetry from AdS 5 × S 5 superstring integrability, Phys. Rev. D 78 (2008) 126004 [arXiv:0807.3228] [SPIRES].MathSciNetADSGoogle Scholar
  84. [84]
    L.J. Mason, Global anti-self-dual Yang-Mills fields in split signature and their scattering, math-ph/0505039 [SPIRES].
  85. [85]
    L.J. Mason, Global solutions of the self-duality equations in split signature, in Further Advances In twistor Theory, volume II, Pitman Research Notes in Maths 232, U.S.A. (1995), pg. 39 [SPIRES].
  86. [86]
    G.A. Sparling, Inversion for the radon line transform in higher dimensions, Phil. Trans. Roy. Soc. Ser. A 356 (1998) 3041.CrossRefMathSciNetADSGoogle Scholar
  87. [87]
    N.M.J. Woodhouse, Contour integrals for the ultra-hyperbolic wave equation, Proc. Roy. Soc. London A 438 (1992) 197.MathSciNetADSGoogle Scholar
  88. [88]
    T.N. Bailey, M.G. Eastwood, R. Gover and L.J. Mason, The Funk transform as a Penrose transform, Math. Proc. Camb. Phil. Soc. 125 (1999) 67.zbMATHCrossRefMathSciNetGoogle Scholar
  89. [89]
    T.N. Bailey and M.G. Eastwood, Twistor results for integral transforms, Contemp. Math. 278 (2001) 77.MathSciNetGoogle Scholar
  90. [90]
    C. LeBrun and L.J. Mason, Zoll manifolds and complex surfaces, J. Diff. Geom. 61 (2002) 453.zbMATHMathSciNetGoogle Scholar
  91. [91]
    C. LeBrun and L.J. Mason, Nonlinear gravitons, null geodesics and holomorphic disks, math/0504582 [SPIRES].
  92. [92]
    C. LeBrun and L.J. Mason, The Einstein-Weyl equations, scattering maps and holomorphic disks, arXiv:0806.3761 [SPIRES].
  93. [93]
    A. Ferber, Supertwistors and conformal supersymmetry, Nucl. Phys. B 132 (1978) 55 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  94. [94]
    I.L. Buchbinder and S.M. Kuzenko, Ideas and methods of supersymmetry and supergravity: or a walk through superspace, IOP, Bristol U.K. (1998).zbMATHGoogle Scholar

Copyright information

© The Author(s) 2010

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

  1. 1.The Mathematical InstituteUniversity of OxfordOxfordU.K.
  2. 2.Institut des Hautes Études Scientifiques, Le Bois MarieBures-sur-YvetteFrance

Personalised recommendations