- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Structure constants from Q systems and separation of variables

Till Bargheer1,*, Carlos Bercini2,†, Gabriel Lefundes3,4,‡, and Paul Ryan1,§

  • *Contact author: till.bargheer@desy.de
  • †Contact author: carlos.bercini@kcl.ac.uk
  • ‡Contact author: g.lefundes@gmail.com
  • §Contact author: paul.ryan@desy.de
PDF

Phys. Rev. D 114, L081904 – Published 8 October, 2026

DOI: https://doi.org/10.1103/tm1j-wxq7

Abstract

We introduce a novel method to compute structure constants from Q functions in the scalar sector of planar N=4 super Yang-Mills (SYM) and related theories. The method derives from operatorial as well as functional separation of variables, and the structure constants are expressed as determinants of matrices whose entries are integrals over products of Q functions. In this framework, each operator is twisted by an external angle, mirroring the cusped Maldacena-Wilson loop. The structure constants of local single-trace operators in N=4 SYM are recovered in the untwisting limit, where we obtain a one-to-one correspondence between our key building blocks and those of the hexagon formalism. Retaining appropriate twists, our structure constants also perfectly match those of the orbifold points of N=4 SYM. Our results thus far are valid at leading order in the weak-coupling expansion, but their formulation in terms of Q functions provides a natural starting point for including loop corrections. Many of the methods we develop in this work apply more generally to the computation of correlation functions in integrable models.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (41)

  1. N. Gromov, V. Kazakov, S. Leurent, and D. Volin, Quantum spectral curve for planar N=4 super-Yang–Mills theory, Phys. Rev. Lett. 112, 011602 (2014).
  2. N. Gromov, V. Kazakov, S. Leurent, and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS5/CFT4, J. High Energy Phys. 09 (2015) 187.
  3. N. Gromov, A. Hegedus, J. Julius, and N. Sokolova, Fast QSC solver: Tool for systematic study of N=4 super-Yang–Mills spectrum, J. High Energy Phys. 05 (2024) 185.
  4. S. Ekhammar, N. Gromov, and M. Preti, Regge trajectories of N=4 SYM. Part I. General asymptotic Baxter-Bethe ansatz, J. High Energy Phys. 02 (2024) 027.
  5. B. Basso, S. Komatsu, and P. Vieira, Structure constants and integrable bootstrap in planar N=4 SYM theory (2015), arXiv:1505.06745.
  6. B. Basso, V. Gonçalves, and S. Komatsu, Structure constants at wrapping order, J. High Energy Phys. 05 (2017) 124.
  7. E. K. Sklyanin, The quantum Toda chain, Lect. Notes Phys. 226, 196 (1985).
  8. S. Giombi and S. Komatsu, Exact correlators on the Wilson loop in N=4 SYM: Localization, defect CFT, and integrability, J. High Energy Phys. 05 (2018) 109.
  9. A. Cavaglià, N. Gromov, and F. Levkovich-Maslyuk, Quantum spectral curve and structure constants in N=4 SYM: Cusps in the ladder limit, J. High Energy Phys. 10 (2018) 060.
  10. Y. Jiang, S. Komatsu, I. Kostov, and D. Serban, The hexagon in the mirror: The three-point function in the SoV representation, J. Phys. A 49, 174007 (2016).
  11. C. Bercini, A. Homrich, and P. Vieira, Structure constants in N=4 supersymmetric Yang–Mills theory and separation of variables, Phys. Rev. D 110, L121901 (2024).
  12. T. Bargheer, C. Bercini, A. Cavaglià, D. Lai, and P. Ryan, Orthogonality of q-functions up to wrapping in planar N=4 super Yang–Mills theory, SciPost Phys. 20, 064 (2026).
  13. N. Gromov, F. Levkovich-Maslyuk, and G. Sizov, New construction of eigenstates and separation of variables for SU(N) quantum spin chains, J. High Energy Phys. 09 (2017) 111.
  14. J. M. Maillet and G. Niccoli, On quantum separation of variables, J. Math. Phys. (N.Y.) 59, 091417 (2018).
  15. P. Ryan and D. Volin, Separated variables and wave functions for rational gl(n) spin chains in the companion twist frame, J. Math. Phys. (N.Y.) 60, 032701 (2019).
  16. A. Cavaglià, N. Gromov, and F. Levkovich-Maslyuk, Separation of variables and scalar products at any rank, J. High Energy Phys. 09 (2019) 052.
  17. N. Gromov, F. Levkovich-Maslyuk, P. Ryan, and D. Volin, Dual separated variables and scalar products, Phys. Lett. B 806, 135494 (2020).
  18. N. Gromov, F. Levkovich-Maslyuk, and P. Ryan, Determinant form of correlators in high rank integrable spin chains via separation of variables, J. High Energy Phys. 05 (2021) 169.
  19. S. Kachru and E. Silverstein, 4-d conformal theories and strings on orbifolds, Phys. Rev. Lett. 80, 4855 (1998).
  20. See Supplemental Material at http://link.aps.org/supplemental/10.1103/tm1j-wxq7, which includes additional derivations, proofs, and checks.
  21. N. Drukker and J. Plefka, Superprotected n-point correlation functions of local operators in N=4 super Yang–Mills, J. High Energy Phys. 04 (2009) 052.
  22. E. Mukhin, V. Tarasov, and A. Varchenko, Spaces of quasi-exponentials and representations of the Yangian Y(glN), Transform. Groups 19, 861 (2014).
  23. D. Chernyak, S. Leurent, and D. Volin, Completeness of Wronskian Bethe equations for rational glm|n spin chains, Commun. Math. Phys. 391, 969 (2022).
  24. C. Kristjansen, D. Müller, and K. Zarembo, Duality relations for overlaps of integrable boundary states in AdS/dCFT, J. High Energy Phys. 09 (2021) 004.
  25. B. Basso, F. Coronado, S. Komatsu, H. T. Lam, P. Vieira, and D.-l. Zhong, Asymptotic four point functions, J. High Energy Phys. 07 (2019) 082.
  26. S. Ekhammar, N. Gromov, and P. Ryan, Boundary overlaps from functional separation of variables, J. High Energy Phys. 05 (2024) 268.
  27. J. Caetano and S. Komatsu, Functional equations and separation of variables for exact g-function, J. High Energy Phys. 09 (2020) 180.
  28. A. Cavaglià, N. Gromov, and F. Levkovich-Maslyuk, Separation of variables in AdS/CFT: Functional approach for the fishnet CFT, J. High Energy Phys. 06 (2021) 131.
  29. N. Beisert and R. Roiban, The Bethe ansatz for ZS orbifolds of N=4 super Yang–Mills theory, J. High Energy Phys. 11 (2005) 037.
  30. T. Skrzypek, Integrability treatment of AdS/CFT orbifolds, J. Phys. A 56, 345401 (2023).
  31. D. le Plat and T. Skrzypek, Three-point functions from integrability in N=2 orbifold theories, J. High Energy Phys. 12 (2025) 172.
  32. G. Ferrando, S. Komatsu, G. Lefundes, and D. Serban, Exact three-point functions in N=2 superconformal field theories: Integrability vs localization, J. High Energy Phys. 05 (2026) 289.
  33. B. Basso, A. Georgoudis, and A. Klemenchuk Sueiro, Structure constants of short operators in planar N=4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 130, 131603 (2023).
  34. B. Basso and A. Georgoudis, Exploring structure constants in planar N=4 SYM: From small spin to strong coupling, J. High Energy Phys. 12 (2025) 119.
  35. R. Pereira, Correlation Functions in Integrable Theories: From weak to strong coupling, Ph.D. thesis, Uppsala University, 2017.
  36. S. Ekhammar and D. Volin, Bethe algebra using pure spinors, Lett. Math. Phys. 115, 20 (2025).
  37. A. Cavaglià, D. Grabner, N. Gromov, and A. Sever, Colour-twist operators. Part I. Spectrum and wave functions, J. High Energy Phys. 06 (2020) 092.
  38. B. Eden, D. le Plat, and A. Spiering, Higher-rank sectors in the hexagon formalism and marginal deformations, J. Phys. A 57, 475401 (2024).
  39. A. M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B164, 171 (1980).
  40. N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve for a cusped Wilson line in N=4 SYM, J. High Energy Phys. 04 (2016) 134.
  41. A. Homrich, D. Simmons-Duffin, and P. Vieira, Light-ray wave functions and integrability, J. High Energy Phys. 10 (2024) 125.

Outline

Information

Phys. Rev. D 114, L081904– Published 8 October, 2026
  • Received 2 April 2026
  • Accepted 27 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation