Skip to main content
SpringerLink
Log in

Scattering into the fifth dimension of \( \mathcal{N} \) = 4 super Yang-Mills

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We study an alternative to dimensional regularisation of planar scattering amplitudes in \( \mathcal{N} \) = 4 super Yang-Mills theory by going to the Coulomb phase of the theory. The infrared divergences are regulated by masses obtained from a Higgs mechanism, allowing us to work in four dimensions. The corresponding string theory set-up suggests that the amplitudes have an exact dual conformal symmetry. The latter acts on the kinematical variables of the amplitudes as well as on the Higgs masses in an effectively five dimensional space. We confirm this expectation by an explicit calculation in the gauge theory. A consequence of this exact dual conformal symmetry is a significantly reduced set of scalar basis integrals that are allowed to appear in an amplitude. For example, triangle sub-graphs are ruled out. We argue that the study of exponentiation of amplitudes is simpler in the Higgsed theory because evanescent terms in the mass regulator can be consistently dropped. We illustrate this by showing the exponentiation of a four-point amplitude to two loops. Finally, we also analytically compute the small mass expansion of a two-loop master integral with an internal mass.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].

    MATH  MathSciNet  ADS  Google Scholar 

  2. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  3. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].

    MATH  MathSciNet  Google Scholar 

  4. J.A. Minahan and K. Zarembo, The Bethe-ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of \( \mathcal{N} \) = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. 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].

    MathSciNet  ADS  Google Scholar 

  7. A.A. Tseytlin, Semiclassical strings in AdS 5 × S 5 and scalar operators in N = 4 SYM theory, Comptes Rendus Physique 5 (2004) 1049 [hep-th/0407218] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  8. A.V. Belitsky, V.M. Braun, A.S. Gorsky and G.P. Korchemsky, Integrability in QCD and beyond, Int. J. Mod. Phys. A 19 (2004) 4715 [hep-th/0407232] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  9. N. Beisert, The dilatation operator of N = 4 super Yang-Mills theory and integrability, Phys. Rept. 405 (2005) 1 [hep-th/0407277] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  10. J. Plefka, Spinning strings and integrable spin chains in the AdS/CFT correspondence, Living Rev. Rel. 8 (2005) 9 [hep-th/0507136] [SPIRES].

    Google Scholar 

  11. J.A. Minahan, A brief introduction to the Bethe ansatz In \( \mathcal{N} \) = 4 super-Yang-Mills, J. Phys. A 39 (2006) 12657 [SPIRES].

    MathSciNet  Google Scholar 

  12. G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  13. S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [SPIRES].

    Article  ADS  Google Scholar 

  14. F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [SPIRES].

    Article  ADS  Google Scholar 

  15. E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. 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].

    Article  MathSciNet  ADS  Google Scholar 

  17. 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].

    Article  MathSciNet  ADS  Google Scholar 

  18. N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, arXiv:0808.1446 [SPIRES].

  19. H. Elvang, D.Z. Freedman and M. Kiermaier, Recursion relations, generating functions, and unitarity sums in N = 4 SYM theory, JHEP 04 (2009) 009 [arXiv:0808.1720] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  20. J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  21. 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].

    Article  MathSciNet  ADS  Google Scholar 

  22. 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] [SPIRES].

    Article  ADS  Google Scholar 

  23. C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. Z. Bern, L.J. Dixon and D.A. Kosower, On-shell methods in perturbative QCD, Annals Phys. 322 (2007) 1587 [arXiv:0704.2798] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  25. 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].

    MathSciNet  ADS  Google Scholar 

  26. 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].

    MathSciNet  ADS  Google Scholar 

  27. C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  28. 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].

    MathSciNet  ADS  Google Scholar 

  29. L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. J. Bartels, L.N. Lipatov and A. Sabio Vera, BFKL Pomeron, reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [SPIRES].

    ADS  Google Scholar 

  31. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, The hexagon Wilson loop and the BDS ansatz for the six- gluon amplitude, Phys. Lett. B 662 (2008) 456 [arXiv:0712.4138] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  32. 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].

    Article  MathSciNet  ADS  Google Scholar 

  33. 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].

    Article  MathSciNet  ADS  Google Scholar 

  34. L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  35. J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  36. 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].

    Article  MathSciNet  ADS  Google Scholar 

  37. 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].

    Article  Google Scholar 

  38. 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].

    Article  Google Scholar 

  39. 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].

    MathSciNet  ADS  Google Scholar 

  40. T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 superconformal symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [SPIRES].

    Article  Google Scholar 

  41. G.P. Korchemsky and E. Sokatchev, Symmetries and analytic properties of scattering amplitudes in N = 4 SYM theory, arXiv:0906.1737 [SPIRES].

  42. N. Berkovits and J. Maldacena, Fermionic T-duality, dual superconformal symmetry and the amplitude/Wilson loop connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  43. 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].

    MathSciNet  ADS  Google Scholar 

  44. N. Beisert, T-duality, dual conformal symmetry and integrability for strings on AdS 5 × S 5, Fortschr. Phys. 57 (2009) 329 [arXiv:0903.0609] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  45. A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  46. L.F. Alday, Lectures on scattering amplitudes via AdS/CFT, Fortsch. Phys. 56 (2008) 816 [arXiv:0804.0951] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  47. L.F. Alday and R. Roiban, Scattering amplitudes, Wilson loops and the string/gauge theory correspondence, Phys. Rept. 468 (2008) 153 [arXiv:0807.1889] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  48. J.M. Henn, Duality between Wilson loops and gluon amplitudes, Fortsch. Phys. 57 (2009) 729 [arXiv:0903.0522] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  49. S.V. Ivanov, G.P. Korchemsky and A.V. Radyushkin, Infrared asymptotics of perturbative QCD: contour gauges, Yad. Fiz. 44 (1986) 230 [Sov. J. Nucl. Phys. 44 (1986) 145] [SPIRES].

    Google Scholar 

  50. Z. Komargodski, On collinear factorization of Wilson loops and MHV amplitudes in N = 4 SYM, JHEP 05 (2008) 019 [arXiv:0801.3274] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  51. A.M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B 164 (1980) 171 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  52. G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [SPIRES].

    Article  ADS  Google Scholar 

  53. I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [SPIRES].

    ADS  Google Scholar 

  54. N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [SPIRES].

  55. R. Ricci, A.A. Tseytlin and M. Wolf, On T-duality and integrability for strings on AdS backgrounds, JHEP 12 (2007) 082 [arXiv:0711.0707] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  56. 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].

    Article  ADS  Google Scholar 

  57. H. Kawai and T. Suyama, Some implications of perturbative approach to AdS/CFT correspondence, Nucl. Phys. B 794 (2008) 1 [arXiv:0708.2463] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  58. R.M. Schabinger, Scattering on the moduli space of N = 4 super Yang-Mills, arXiv:0801.1542 [SPIRES].

  59. J. McGreevy and A. Sever, Planar scattering amplitudes from Wilson loops, JHEP 08 (2008) 078 [arXiv:0806.0668] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  60. G. ’t Hooft and M.J.G. Veltman, Scalar one loop integrals, Nucl. Phys. B 153 (1979) 365 [SPIRES].

    Article  ADS  Google Scholar 

  61. A. Brandhuber, P. Heslop and G. Travaglini, One-loop amplitudes in N = 4 super Yang-Mills and anomalous dual conformal symmetry, JHEP 08 (2009) 095 [arXiv:0905.4377] [SPIRES].

    Article  Google Scholar 

  62. H. Elvang, D.Z. Freedman and M. Kiermaier, Dual conformal symmetry of 1-loop NMHV amplitudes in N = 4 SYM theory, arXiv:0905.4379 [SPIRES].

  63. A. Brandhuber, P. Heslop and G. Travaglini, Proof of the dual conformal anomaly of one-loop amplitudes in N = 4 SYM, JHEP 10 (2009) 063 [arXiv:0906.3552] [SPIRES].

    Article  Google Scholar 

  64. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Generalized unitarity for N = 4 super-amplitudes, arXiv:0808.0491 [SPIRES].

  65. Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [SPIRES].

    Article  ADS  Google Scholar 

  66. J. McGreevy and A. Sever, Quark scattering amplitudes at strong coupling, JHEP 02 (2008) 015 [arXiv:0710.0393] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  67. M. Kruczenski, R. Roiban, A. Tirziu and A.A. Tseytlin, Strong-coupling expansion of cusp anomaly and gluon amplitudes from quantum open strings in AdS 5 × S 5, Nucl. Phys. B 791 (2008) 93 [arXiv:0707.4254] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  68. V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1 [SPIRES].

    Google Scholar 

  69. J. Gluza, K. Kajda and T. Riemann, AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals, Comput. Phys. Commun. 177 (2007) 879 [arXiv:0704.2423] [SPIRES].

    Article  ADS  Google Scholar 

  70. L.F. Alday and J. Maldacena, Minimal surfaces in AdS and the eight-gluon scattering amplitude at strong coupling, arXiv:0903.4707 [SPIRES].

  71. L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, arXiv:0904.0663 [SPIRES].

  72. A. Jevicki, K. Jin, C. Kalousios and A. Volovich, Generating AdS string solutions, JHEP 03 (2008) 032 [arXiv:0712.1193] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, U.S.A.

    Luis F. Alday

  2. Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489, Berlin, Germany

    Johannes Henn, Jan Plefka & Theodor Schuster

Authors
  1. Luis F. Alday
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johannes Henn
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Plefka
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Theodor Schuster
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis F. Alday.

Additional information

ArXiv ePrint: 0908.0684

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alday, L.F., Henn, J., Plefka, J. et al. Scattering into the fifth dimension of \( \mathcal{N} \) = 4 super Yang-Mills. J. High Energ. Phys. 2010, 77 (2010). https://doi.org/10.1007/JHEP01(2010)077

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP01(2010)077

Keywords

Search

Navigation