Skip to main content
SpringerLink
Log in

Analytic two-loop form factors in \( \mathcal{N} = 4 \) SYM

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

Abstract

We derive a compact expression for the three-point MHV form factors of half- BPS operators in \( \mathcal{N} = 4 \) super Yang-Mills at two loops. The main tools of our calculation are generalised unitarity applied at the form factor level, and the compact expressions for supersymmetric tree-level form factors and amplitudes entering the cuts. We confirm that infrared divergences exponentiate as expected, and that collinear factorisation is entirely captured by an ABDK/BDS ansatz. Next, we construct the two-loop remainder function obtained by subtracting this ansatz from the full two-loop form factor and compute it numerically. Using symbology, combined with various physical constraints and symme- tries, we find a unique solution for its symbol. With this input we construct a remarkably compact analytic expression for the remainder function, which contains only classical poly- logarithms, and compare it to our numerical results. Furthermore, we make the surprising observation that our remainder is equal to the maximally transcendental piece of the two- loop Higgs plus three-gluon scattering amplitudes in QCD.

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

    Article  MathSciNet  ADS  Google Scholar 

  2. 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] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  3. N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].

    Article  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  5. J. Maldacena and A. Zhiboedov, Form factors at strong coupling via a Y-system, JHEP 11 (2010) 104 [arXiv:1009.1139] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. A. Brandhuber, B. Spence, G. Travaglini and G. Yang, Form Factors in N = 4 Super Yang-Mills and Periodic Wilson Loops, JHEP 01 (2011) 134 [arXiv:1011.1899] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini and G. Yang, Harmony of Super Form Factors, JHEP 10 (2011) 046 [arXiv:1107.5067] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. W. van Neerven, Infrared Behavior Of On-Shell Form-Factors In A N = 4 Supersymmetric Yang-Mills Field Theory, Z. Phys. C 30 (1986) 595 [INSPIRE].

    ADS  Google Scholar 

  9. L. Bork, D. Kazakov and G. Vartanov, On form factors in N = 4 SYM, JHEP 02 (2011) 063 [arXiv:1011.2440] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. L. Bork, D. Kazakov and G. Vartanov, On MHV Form Factors in Superspace for \ = 4 SYM Theory, JHEP 10 (2011) 133 [arXiv:1107.5551] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. C.R. Schmidt, H → ggg \( \left( {gq\overline q } \right) \) at two loops in the large M(t) limit, Phys. Lett. B 413 (1997) 391 [hep-ph/9707448] [INSPIRE].

    ADS  Google Scholar 

  12. T. Gehrmann, M. Jaquier, E. Glover and A. Koukoutsakis, Two-Loop QCD Corrections to the Helicity Amplitudes for H → 3 partons, JHEP 02 (2012) 056 [arXiv:1112.3554] [INSPIRE].

    Article  ADS  Google Scholar 

  13. A. Koukoutsakis, Higgs Boson and QCD Jets at Two Loops, PhD thesis, University of Durham (2003).

  14. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. 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] [INSPIRE].

    Article  ADS  Google Scholar 

  16. Z. Bern, L.J. Dixon and D.A. Kosower, Two-loop g → gg splitting amplitudes in QCD, JHEP 08 (2004) 012 [hep-ph/0404293] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. T. Gehrmann and E. Remiddi, Two loop master integrals for γ∗ → 3 jets: The Planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].

    Article  ADS  Google Scholar 

  20. T. Gehrmann and E. Remiddi, Two loop master integrals for γ∗ → 3 jets: The Nonplanar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].

    Article  ADS  Google Scholar 

  21. V.A. Smirnov, Feynman integral calculus, Springer, Berlin Germany (2006),

    Google Scholar 

  22. M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  23. A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. V. Del Duca, C. Duhr and V.A. Smirnov, The Two-Loop Hexagon Wilson Loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  27. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. 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] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  29. J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].

    Article  ADS  Google Scholar 

  31. S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].

    Article  ADS  Google Scholar 

  32. L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. P. Heslop and V.V. Khoze, Wilson Loops @ 3-Loops in Special Kinematics, JHEP 11 (2011) 152 [arXiv:1109.0058] [INSPIRE].

    Article  ADS  Google Scholar 

  34. L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].

    Article  ADS  Google Scholar 

  35. A. Prygarin, M. Spradlin, C. Vergu and A. Volovich, All Two-Loop MHV Amplitudes in Multi-Regge Kinematics From Applied Symbology, arXiv:1112.6365 [INSPIRE].

  36. S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].

    ADS  Google Scholar 

  37. A. Kotikov, L. Lipatov, A. Onishchenko and V. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754-756] [hep-th/0404092] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  38. J.M. Henn, S. Moch and S.G. Naculich, Form factors and scattering amplitudes in N = 4 SYM in dimensional and massive regularizations, JHEP 12 (2011) 024 [arXiv:1109.5057] [INSPIRE].

    Article  ADS  Google Scholar 

  39. C. Boucher-Veronneau and L.J. Dixon, unpublished notes.

  40. T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in N = 4 super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].

    Article  ADS  Google Scholar 

  41. R.J. Gonsalves, Dimensionally regularized two loop on-shell quark form-factor, Phys. Rev. D 28 (1983) 1542 [INSPIRE].

    ADS  Google Scholar 

  42. W. van Neerven, Dimensional regularization of mass and infrared singularities in two loop on-shell vertex functions, Nucl. Phys. B 268 (1986) 453 [INSPIRE].

    Article  ADS  Google Scholar 

  43. G. Kramer and B. Lampe, Integrals for two loop calculations in massless QCD, J. Math. Phys. 28 (1987) 945 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. T. Gehrmann, T. Huber and D. Maître, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [hep-ph/0507061] [INSPIRE].

    ADS  Google Scholar 

  45. D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].

    Article  ADS  Google Scholar 

  46. G. Georgiou, E.N. Glover and V.V. Khoze, Non-MHV tree amplitudes in gauge theory, JHEP 07 (2004) 048 [hep-th/0407027] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. 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] [INSPIRE].

    Article  ADS  Google Scholar 

  49. V. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].

    ADS  Google Scholar 

  50. 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] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. J. Gluza, K. Kajda, T. Riemann and V. Yundin, Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics, Eur. Phys. J. C 71 (2011) 1516 [arXiv:1010.1667] [INSPIRE].

    ADS  Google Scholar 

  52. Z. Bern and G. Chalmers, Factorization in one loop gauge theory, Nucl. Phys. B 447 (1995) 465 [hep-ph/9503236] [INSPIRE].

    Article  ADS  Google Scholar 

  53. D.A. Kosower and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [INSPIRE].

    Article  ADS  Google Scholar 

  54. Z. Bern, V. Del Duca, W.B. Kilgore and C.R. Schmidt, The Infrared behavior of one loop QCD amplitudes at next-to-next-to leading order, Phys. Rev. D 60 (1999) 116001 [hep-ph/9903516] [INSPIRE].

    ADS  Google Scholar 

  55. A. Brandhuber, B. Spence and G. Travaglini, From trees to loops and back, JHEP 01 (2006) 142 [hep-th/0510253] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. J. Drummond, J. Henn, G. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  57. J. Drummond, J. Henn, G. 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].

    Article  MathSciNet  ADS  Google Scholar 

  58. L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [INSPIRE].

    ADS  Google Scholar 

  59. G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].

    ADS  Google Scholar 

  60. T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [INSPIRE].

    Article  ADS  Google Scholar 

  61. E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].

    Article  ADS  Google Scholar 

  62. C. Anastasiou, A. Brandhuber, P. Heslop, V.V. Khoze, B. Spence, et al., Two-Loop Polygon Wilson Loops in N = 4 SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  63. A.B. Goncharov, Polylogarithms and motivic galois groups, Proc. Symp. Pure Math. 55 (1994) 43.

    MathSciNet  Google Scholar 

  64. E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

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

  66. L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  67. D. Zagier, The Dilogarithm Function, Les Houches lecture notes available at http://maths.dur.ac.uk/∼dma0hg/dilog.pdf.

  68. A. von Manteuffel, An analytical solution for a non-planar massive double box diagram, talk given at ACAT 2011, http://goo.gl/K1Kvl.

Download references

Author information

Authors and Affiliations

  1. Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom

    Andreas Brandhuber & Gabriele Travaglini

  2. Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Andreas Brandhuber

  3. II. Institut fu¨r Theoretische Physik, Universita¨t Hamburg, Luruper Chaussee 149, D-22761, Hamburg, Germany

    Gang Yang

Authors
  1. Andreas Brandhuber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriele Travaglini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Brandhuber.

Additional information

ArXiv ePrint: 1201.4170

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brandhuber, A., Travaglini, G. & Yang, G. Analytic two-loop form factors in \( \mathcal{N} = 4 \) SYM. J. High Energ. Phys. 2012, 82 (2012). https://doi.org/10.1007/JHEP05(2012)082

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP05(2012)082

Keywords

Search

Navigation