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All tree-level MHV form factors in \( \mathcal{N} \) = 4 SYM from twistor space

  • Laura Koster
  • Vladimir Mitev
  • Matthias Staudacher
  • Matthias WilhelmEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We incorporate all gauge-invariant local composite operators into the twistor-space formulation of \( \mathcal{N} \) = 4 SYM theory, detailing and expanding on ideas we presented recently in [1]. The vertices for these operators contain infinitely many terms and we show how they can be constructed by taking suitable derivatives of a light-like Wilson loop in twistor space and shrinking it down to a point. In particular, these vertices directly yield the tree-level MHV super form factors of all composite operators in \( \mathcal{N} \) = 4 SYM theory.

Keywords

Scattering Amplitudes Wilson ’t Hooft and Polyakov loops AdS-CFT Correspondence Supersymmetric gauge theory 

Notes

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.

References

  1. [1]
    L. Koster, V. Mitev, M. Staudacher and M. Wilhelm, Composite operators in the twistor formulation of \( \mathcal{N} \) = 4 SYM theory, arXiv:1603.04471 [INSPIRE].
  2. [2]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    H. Elvang and Y.-t. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].
  4. [4]
    J.M. Henn and J.C. Plefka, Scattering amplitudes in gauge theories, Lect. Notes Phys. 883 (2014) 1 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    R. Boels, L.J. Mason and D. Skinner, Supersymmetric gauge theories in twistor space, JHEP 02 (2007) 014 [hep-th/0604040] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    T. Adamo, M. Bullimore, L. Mason and D. Skinner, Scattering amplitudes and Wilson loops in twistor space, J. Phys. A 44 (2011) 454008 [arXiv:1104.2890] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    R. Boels, L.J. Mason and D. Skinner, From twistor actions to MHV diagrams, Phys. Lett. B 648 (2007) 90 [hep-th/0702035] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Adamo and L. Mason, MHV diagrams in twistor space and the twistor action, Phys. Rev. D 86 (2012) 065019 [arXiv:1103.1352] [INSPIRE].ADSGoogle Scholar
  9. [9]
    L.J. Mason and D. Skinner, The complete planar S-matrix of \( \mathcal{N} \) = 4 SYM as a Wilson loop in twistor space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Bullimore and D. Skinner, Holomorphic linking, loop equations and scattering amplitudes in twistor space, arXiv:1101.1329 [INSPIRE].
  11. [11]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    M. Bullimore, L.J. Mason and D. Skinner, MHV diagrams in momentum twistor space, JHEP 12 (2010) 032 [arXiv:1009.1854] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A.V. Belitsky, G.P. Korchemsky and E. Sokatchev, Are scattering amplitudes dual to super Wilson loops?, Nucl. Phys. B 855 (2012) 333 [arXiv:1103.3008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    L. Koster, V. Mitev and M. Staudacher, A twistorial approach to integrability in \( \mathcal{N} \) = 4 SYM, Fortsch. Phys. 63 (2015) 142 [arXiv:1410.6310] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    D. Chicherin et al., Correlation functions of the chiral stress-tensor multiplet in \( \mathcal{N} \) = 4 SYM, JHEP 06 (2015) 198 [arXiv:1412.8718] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    W.L. van Neerven, Infrared behavior of on-shell form-factors in a \( \mathcal{N} \) = 4 supersymmetric Yang-Mills field theory, Z. Phys. C 30 (1986) 595 [INSPIRE].ADSGoogle Scholar
  19. [19]
    A. Brandhuber, B. Spence, G. Travaglini and G. Yang, Form factors in \( \mathcal{N} \) = 4 super Yang-Mills and periodic Wilson loops, JHEP 01 (2011) 134 [arXiv:1011.1899] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    L.V. Bork, D.I. Kazakov and G.S. Vartanov, On form factors in \( \mathcal{N} \) = 4 SYM, JHEP 02 (2011) 063 [arXiv:1011.2440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini and G. Yang, Harmony of super form factors, JHEP 10 (2011) 046 [arXiv:1107.5067] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    L.V. Bork, D.I. Kazakov and G.S. Vartanov, On MHV form factors in superspace for \( \mathcal{N} \) = 4 SYM theory, JHEP 10 (2011) 133 [arXiv:1107.5551] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J.M. Henn, S. Moch and S.G. Naculich, Form factors and scattering amplitudes in \( \mathcal{N} \) = 4 SYM in dimensional and massive regularizations, JHEP 12 (2011) 024 [arXiv:1109.5057] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in \( \mathcal{N} \) = 4 super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in \( \mathcal{N} \) = 4 SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    L.V. Bork, On NMHV form factors in \( \mathcal{N} \) = 4 SYM theory from generalized unitarity, JHEP 01 (2013) 049 [arXiv:1203.2596] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    O.T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP 03 (2013) 172 [arXiv:1209.0227] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    H. Johansson, D.A. Kosower and K.J. Larsen, Two-loop maximal unitarity with external masses, Phys. Rev. D 87 (2013) 025030 [arXiv:1208.1754] [INSPIRE].ADSGoogle Scholar
  29. [29]
    R.H. Boels, B.A. Kniehl, O.V. Tarasov and G. Yang, Color-kinematic duality for form factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    B. Penante, B. Spence, G. Travaglini and C. Wen, On super form factors of half-BPS operators in \( \mathcal{N} \) = 4 super Yang-Mills, JHEP 04 (2014) 083 [arXiv:1402.1300] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A. Brandhuber, B. Penante, G. Travaglini and C. Wen, The last of the simple remainders, JHEP 08 (2014) 100 [arXiv:1406.1443] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    L.V. Bork, On form factors in \( \mathcal{N} \) = 4 SYM theory and polytopes, JHEP 12 (2014) 111 [arXiv:1407.5568] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    M. Wilhelm, Amplitudes, form factors and the dilatation operator in \( \mathcal{N} \) = 4 SYM theory, JHEP 02 (2015) 149 [arXiv:1410.6309] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    D. Nandan, C. Sieg, M. Wilhelm and G. Yang, Cutting through form factors and cross sections of non-protected operators in \( \mathcal{N} \) = 4 SYM, JHEP 06 (2015) 156 [arXiv:1410.8485] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    F. Loebbert, D. Nandan, C. Sieg, M. Wilhelm and G. Yang, On-shell methods for the two-loop dilatation operator and finite remainders, JHEP 10 (2015) 012 [arXiv:1504.06323] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    L.V. Bork and A.I. Onishchenko, On soft theorems and form factors in \( \mathcal{N} \) = 4 SYM theory, JHEP 12 (2015) 030 [arXiv:1506.07551] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    R. Frassek, D. Meidinger, D. Nandan and M. Wilhelm, On-shell diagrams, Graßmannians and integrability for form factors, JHEP 01 (2016) 182 [arXiv:1506.08192] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    R. Boels, B.A. Kniehl and G. Yang, Master integrals for the four-loop Sudakov form factor, Nucl. Phys. B 902 (2016) 387 [arXiv:1508.03717] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    R. Huang, Q. Jin and B. Feng, Form factor and boundary contribution of amplitude, JHEP 06 (2016) 072 [arXiv:1601.06612] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J. Maldacena and A. Zhiboedov, Form factors at strong coupling via a Y-system, JHEP 11 (2010) 104 [arXiv:1009.1139] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    Z. Gao and G. Yang, Y-system for form factors at strong coupling in AdS 5 and with multi-operator insertions in AdS 3, JHEP 06 (2013) 105 [arXiv:1303.2668] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    M. Wilhelm, Form factors and the dilatation operator in \( \mathcal{N} \) = 4 super Yang-Mills theory and its deformations, arXiv:1603.01145 [INSPIRE].
  44. [44]
    J.M. Drummond and J.M. Henn, All tree-level amplitudes in \( \mathcal{N} \) = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar \( \mathcal{N} \) = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    N. Beisert, The dilatation operator of \( \mathcal{N} \) = 4 super Yang-Mills theory and integrability, Phys. Rept. 405 (2004) 1 [hep-th/0407277] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    J.A. Minahan, Review of AdS/CFT integrability, chapter I.1: spin chains in \( \mathcal{N} \) = 4 super Yang-Mills, Lett. Math. Phys. 99 (2012) 33 [arXiv:1012.3983] [INSPIRE].
  50. [50]
    T. Adamo, M. Bullimore, L. Mason and D. Skinner, A proof of the supersymmetric correlation function/Wilson loop correspondence, JHEP 08 (2011) 076 [arXiv:1103.4119] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    L. Koster, V. Mitev, M. Staudacher and M. Wilhelm, to appear.Google Scholar
  52. [52]
    D. Chicherin and E. Sokatchev, \( \mathcal{N} \) = 4 super-Yang-Mills in LHC superspace. Part I: classical and quantum theory, arXiv:1601.06803 [INSPIRE].
  53. [53]
    D. Chicherin and E. Sokatchev, \( \mathcal{N} \) = 4 super-Yang-Mills in LHC superspace. Part II: non-chiral correlation functions of the stress-tensor multiplet, arXiv:1601.06804 [INSPIRE].
  54. [54]
    D. Chicherin and E. Sokatchev, Demystifying the twistor construction of composite operators in \( \mathcal{N} \) = 4 super-Yang-Mills theory, arXiv:1603.08478 [INSPIRE].
  55. [55]
    S. Derkachov, G.P. Korchemsky and A.N. Manashov, Dual conformal symmetry on the light-cone, Nucl. Phys. B 886 (2014) 1102 [arXiv:1306.5951] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    T. Adamo, Twistor actions for gauge theory and gravity, arXiv:1308.2820 [INSPIRE].
  57. [57]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Laura Koster
    • 1
  • Vladimir Mitev
    • 1
    • 2
  • Matthias Staudacher
    • 1
  • Matthias Wilhelm
    • 1
    • 3
    Email author
  1. 1.Institut für Mathematik, Institut für Physik und IRIS AdlershofHumboldt-Universität zu BerlinBerlinGermany
  2. 2.PRISMA Cluster of Excellence, Institut für Physik, WA THEPJohannes Gutenberg-Universität MainzMainzGermany
  3. 3.Niels Bohr InstituteCopenhagen UniversityCopenhagenDenmark

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