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Nonplanar four-point integrand and Konishi dimension in N=4 super-Yang-Mills theory at five loops

Till Bargheer* and Albert Bekov†

  • *Contact author: till.bargheer@desy.de
  • †Contact author: albert.bekov@desy.de
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Phys. Rev. D 114, 045008 – Published 11 August, 2026

DOI: https://doi.org/10.1103/l9hj-dmf1

Abstract

We compute the complete nonplanar integrand for the correlation function of four lightest scalar operators in N=4 super Yang-Mills theory at five-loop order. This is equivalent to the supercorrelator of nine stress-tensor multiplets in the self-dual theory. Starting with an ansatz of f graphs, we impose constraints from light-cone limits, and fix the remaining freedom by using the reformulation of the theory in twistor space. We develop an efficient GPU-based algorithm for the numerical evaluation of the twistor rules. Our result for the integrand suggests a sharpened f-graph genus conjecture. As an application, we extract the five-loop nonplanar anomalous dimension of the Konishi operator, whose 1/Nc2 term is entirely given in terms of odd zeta values of transcendentality five and higher. Our code and result are provided in ancillary files.

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References (67)

  1. F. Gonzalez-Rey, I. Y. Park, and K. Schalm, A note on four point functions of conformal operators in N=4 superYang–Mills, Phys. Lett. B 448, 37 (1999).
  2. B. Eden, P. S. Howe, C. Schubert, E. Sokatchev, and P. C. West, Four point functions in N=4 supersymmetric Yang-Mills theory at two loops, Nucl. Phys. B557, 355 (1999).
  3. B. Eden, P. S. Howe, C. Schubert, E. Sokatchev, and P. C. West, Simplifications of four point functions in N=4 supersymmetric Yang-Mills theory at two loops, Phys. Lett. B 466, 20 (1999).
  4. B. Eden, C. Schubert, and E. Sokatchev, Three loop four point correlator in N=4 SYM, Phys. Lett. B 482, 309 (2000).
  5. M. Bianchi, S. Kovacs, G. Rossi, and Y. S. Stanev, Anomalous dimensions in N=4 SYM theory at order g**4, Nucl. Phys. B584, 216 (2000).
  6. B. Eden, P. Heslop, G. P. Korchemsky, and E. Sokatchev, Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM, Nucl. Phys. B862, 193 (2012).
  7. B. Eden, P. Heslop, G. P. Korchemsky, and E. Sokatchev, Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM, Nucl. Phys. B862, 450 (2012).
  8. R. G. Ambrosio, B. Eden, T. Goddard, P. Heslop, and C. Taylor, Local integrands for the five-point amplitude in planar N=4 SYM up to five loops, J. High Energy Phys. 01 (2013) 116.
  9. J. L. Bourjaily, P. Heslop, and V.-V. Tran, Perturbation theory at eight loops: Novel structures and the breakdown of manifest conformality in N=4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 116, 191602 (2016).
  10. J. L. Bourjaily, P. Heslop, and V.-V. Tran, Amplitudes and correlators to ten loops using simple, graphical bootstraps, J. High Energy Phys. 11 (2016) 125.
  11. S. He, C. Shi, Y. Tang, and Y.-Q. Zhang, The cusp limit of correlators and a new graphical bootstrap for correlators/amplitudes to eleven loops, J. High Energy Phys. 03 (2025) 192.
  12. J. L. Bourjaily, S. He, C. Shi, and Y. Tang, The four-point correlator of planar sym at twelve loops, arXiv:2503.15593.
  13. E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B562, 353 (1999).
  14. E. D’Hoker, D. Z. Freedman, and L. Rastelli, AdS/CFT four point functions: How to succeed at z integrals without really trying, Nucl. Phys. B562, 395 (1999).
  15. G. Arutyunov and S. Frolov, Four point functions of lowest weight cpos in N=4 SYM(4) in supergravity approximation, Phys. Rev. D 62, 064016 (2000).
  16. L. Rastelli and X. Zhou, Mellin amplitudes for supergravity on AdS5×S5, Phys. Rev. Lett. 118, 091602 (2017).
  17. L. F. Alday and A. Bissi, Loop corrections to supergravity on AdS5×S5, Phys. Rev. Lett. 119, 171601 (2017).
  18. F. Aprile, J. M. Drummond, P. Heslop, and H. Paul, Quantum gravity from conformal field theory, J. High Energy Phys. 01 (2017) 035.
  19. L. Rastelli and X. Zhou, How to succeed at holographic correlators without really trying, J. High Energy Phys. 04 (2017) 014.
  20. E. Y. Yuan, Simplicity in AdS perturbative dynamics, arXiv:1801.07283.
  21. D. Carmi, Loops in AdS: From the spectral representation to position space, J. High Energy Phys. 06 (2019) 049.
  22. J. M. Drummond and H. Paul, One-loop string corrections to AdS amplitudes from CFT, J. High Energy Phys. 03 (2019) 038.
  23. Z. Huang and E. Y. Yuan, Graviton scattering in AdS5×S5 at two loops, J. High Energy Phys. 04 (2021) 064.
  24. J. M. Drummond and H. Paul, Two-loop supergravity on AdS5×S5 from CFT, J. High Energy Phys. 08 (2022) 275.
  25. S. Caron-Huot, F. Coronado, A.-K. Trinh, and Z. Zahraee, Bootstrapping N=4 sym correlators using integrability, J. High Energy Phys. 02 (2022) 083.
  26. Z. Huang, B. Wang, and E. Y. Yuan, A differential representation for holographic correlators, J. High Energy Phys. 07 (2024) 176.
  27. S. Caron-Huot, F. Coronado, and Z. Zahraee, Bootstrapping N=4 sym correlators using integrability and localization, J. High Energy Phys. 05 (2024) 220.
  28. A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, From correlation functions to event shapes, Nucl. Phys. B884, 305 (2014).
  29. A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, Event shapes in N=4 super-Yang-Mills theory, Nucl. Phys. B884, 206 (2014).
  30. A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, Energy-energy correlations in N=4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 112, 071601 (2014).
  31. K. A. Intriligator, Bonus symmetries of N=4 super-Yang-Mills correlation functions via AdS duality, Nucl. Phys. B551, 575 (1999).
  32. B. Eden, P. S. Howe, C. Schubert, E. Sokatchev, and P. C. West, Extremal correlators in four-dimensional SCFT, Phys. Lett. B 472, 323 (2000).
  33. T. Fleury and R. Pereira, Non-planar data of N=4 SYM, J. High Energy Phys. 03 (2019) 003.
  34. B. Eden, A. C. Petkou, C. Schubert, and E. Sokatchev, Partial non-renormalisation of the stress-tensor four-point function in N=4 SYM4 and AdS/CFT, Nucl. Phys. B607, 191 (2001).
  35. G. Brinkmann, A practical algorithm for the computation of the genus, ARS Math. Contemp. 22, 4 (2022).
  36. L. F. Alday, B. Eden, G. P. Korchemsky, J. Maldacena, and E. Sokatchev, From correlation functions to Wilson loops, J. High Energy Phys. 09 (2011) 123.
  37. The Sage Developers, SageMath, the Sage Mathematics Software System (Version 10.7) (2025), https://www.sagemath.org.
  38. T. Bargheer and A. Bekov, The non-planar four-point integrand and Konishi dimension in N=4 super Yang-Mills theory at five loops, arXiv:2511.21807.
  39. B. Eden, P. Heslop, G. P. Korchemsky, V. A. Smirnov, and E. Sokatchev, Five-loop Konishi in N=4 SYM, Nucl. Phys. B862, 123 (2012).
  40. A. V. Smirnov and F. S. Chukharev, FIRE6: Feynman Integral REduction with modular arithmetic, Comput. Phys. Commun. 247, 106877 (2020).
  41. A. V. Smirnov and M. Zeng, FIRE 6.5: Feynman integral reduction with new simplification library, Comput. Phys. Commun. 302, 109261 (2024).
  42. P. A. Baikov and K. G. Chetyrkin, Four loop massless propagators: An algebraic evaluation of all master integrals, Nucl. Phys. B837, 186 (2010).
  43. R. N. Lee, A. V. Smirnov, and V. A. Smirnov, Master integrals for four-loop massless propagators up to transcendentality weight twelve, Nucl. Phys. B856, 95 (2012).
  44. D. J. Broadhurst and D. Kreimer, Knots and numbers in phi**4 theory to 7 loops and beyond, Int. J. Mod. Phys. C 06, 519 (1995).
  45. F. Brown and O. Schnetz, Proof of the zig-zag conjecture, arXiv:1208.1890.
  46. S. Leurent and D. Volin, Multiple zeta functions and double wrapping in planar N=4 SYM, Nucl. Phys. B875, 757 (2013).
  47. V. N. Velizhanin, The non-planar contribution to the four-loop universal anomalous dimension in N=4 supersymmetric Yang-Mills theory, JETP Lett. 89, 593 (2009).
  48. S.-Q. Zhang, Nonplanar integrated correlator in N=4 SYM, Phys. Rev. D 110, 025003 (2024).
  49. D. Dorigoni, M. B. Green, and C. Wen, Novel representation of an integrated correlator in N=4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 126, 161601 (2021).
  50. D. Dorigoni, M. B. Green, and C. Wen, Exact properties of an integrated correlator in N=4 su(n) SYM, J. High Energy Phys. 05 (2021) 089.
  51. A. Georgoudis, V. Gonçalves, and R. Pereira, Konishi ope coefficient at the five loop order, J. High Energy Phys. 11 (2017) 184.
  52. J. M. Henn, G. P. Korchemsky, and B. Mistlberger, The full four-loop cusp anomalous dimension in N=4 super Yang-Mills and QCD, J. High Energy Phys. 04 (2019) 018,
  53. D. Anselmi, M. T. Grisaru, and A. Johansen, A critical behavior of anomalous currents, electric—magnetic universality and CFT in four-dimensions, Nucl. Phys. B491, 221 (1997).
  54. A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko, and V. N. Velizhanin, Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang-Mills model, Phys. Lett. B 595, 521 (2004).
  55. F. Fiamberti, A. Santambrogio, C. Sieg, and D. Zanon, Anomalous dimension with wrapping at four loops in N=4 SYM, Nucl. Phys. B805, 231 (2008).
  56. V. N. Velizhanin, The four-loop anomalous dimension of the Konishi operator in N=4 supersymmetric Yang-Mills theory, JETP Lett. 89, 6 (2009).
  57. Z. Bajnok and R. A. Janik, Four-loop perturbative konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B807, 625 (2009).
  58. Z. Bajnok, A. Hegedus, R. A. Janik, and T. Lukowski, Five loop Konishi from AdS/CFT, Nucl. Phys. B827, 426 (2010).
  59. G. Arutyunov, S. Frolov, and R. Suzuki, Five-loop konishi from the mirror TBA, J. High Energy Phys. 04 (2010) 069.
  60. J. Balog and A. Hegedus, 5-loop Konishi from linearized TBA and the xxx magnet, J. High Energy Phys. 06 (2010) 080.
  61. N. Beisert, B. Eden, and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 07, P01021 (2007). hep-th/0610251.
  62. R. Boels, L. J. Mason, and D. Skinner, Supersymmetric gauge theories in twistor space, J. High Energy Phys. 02 (2006) 014.
  63. D. Chicherin, R. Doobary, B. Eden, P. Heslop, G. P. Korchemsky, L. Mason, and E. Sokatchev, Correlation functions of the chiral stress-tensor multiplet in N=4 SYM, J. High Energy Phys. 06 (2014) 198.
  64. S. K. Lam, A. Pitrou, and S. Seibert, Numba: A LLVM-based Python JIT compiler, in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, LLVM ’15 (Association for Computing Machinery, New York, USA, 2015).
  65. E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188, 148 (2015).
  66. A. V. Smirnov, N. D. Shapurov, and L. I. Vysotsky, FIESTA5: Numerical high-performance Feynman integral evaluation, Comput. Phys. Commun. 277, 108386 (2022).
  67. O. Schnetz, HyperlogProcedures, version 0.8 (2025).

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Phys. Rev. D 114, 045008– Published 11 August, 2026
  • Received 12 December 2025
  • Accepted 20 July 2026
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