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  • Open Access

Unitarity Flow Conjecture: An On-Shell Approach to the Renormalization Group

Ameya Chavda, Daniel McLoughlin, Sebastian Mizera, and John Staunton

  • Center for Theoretical Physics, Department of Physics, Columbia University, Pupin Hall, 538 West 120th Street, New York, New York 10027, USA

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Phys. Rev. Lett. 136, 091604 – Published 4 March, 2026

DOI: https://doi.org/10.1103/jy8h-zq9t

Abstract

We propose that the broad architecture of the renormalization group flow in quantum field theories is, at least in part, fixed by unitarity. The precise statement is summarized in the unitarity flow conjecture, which states that the nonlinear S-matrix identities obtained by imposing unitarity imply those needed to derive the renormalization group equations. As a proof of principle, we verify this conjecture to all loops at the leading and subleading logarithmic order in the four-dimensional massless λϕ4 theory using on-shell techniques, without reference to any counterterms or Feynman diagrams.

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References (55)

  1. J. L. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, England, 1996).
  2. J. Zinn-Justin, Phase Transitions and Renormalization Group (Oxford University Press, New York, 2007), 07.
  3. L. Adzhemyan, N. Antonov, and A. Vasiliev, Field Theoretic Renormalization Group in Fully Developed Turbulence (Taylor & Francis, London, 1999).
  4. V. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, Oxford, 2005).
  5. J. C. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, Cambridge Monographs on Mathematical Physics Vol. 26 (Cambridge University Press, Cambridge, England, 1984).
  6. C. G. Callan and Jr., Broken scale invariance in scalar field theory, Phys. Rev. D 2, 1541 (1970).
  7. K. Symanzik, Small distance behavior in field theory and power counting, Commun. Math. Phys. 18, 227 (1970).
  8. K. Symanzik, Small distance behavior analysis and Wilson expansion, Commun. Math. Phys. 23, 49 (1971).
  9. M. V. Kompaniets and E. Panzer, Minimally subtracted six loop renormalization of O(n)-symmetric ϕ4 theory and critical exponents, Phys. Rev. D 96, 036016 (2017).
  10. M. V. Kompaniets, A. Kudlis, and A. I. Sokolov, Critical behavior of the weakly disordered Ising model: Six-loop ε expansion study, Phys. Rev. E 103, 022134 (2021).
  11. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier et al., The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  12. O. Schnetz, ϕ4 theory at seven loops, Phys. Rev. D 107, 036002 (2023).
  13. A. V. Bednyakov, A. S. Fedoruk, and D. I. Kazakov, On the renormalization-group analysis of the SM: Loops, uncertainties, and vacuum stability, arXiv:2509.03369.
  14. M. Borinsky and A. Favorito, Feynman integrals at large loop order and the log-Γ distribution, J. High Energy Phys. 09 (2025) 182.
  15. G. Kovyrshin, N. Meshcheriakov, V. Shatalova, and K. Stepanyantz, Structure of renormalization constants for theories with multiple couplings in the MS-like subtraction schemes, Nucl. Phys. B1022, 117272 (2026).
  16. R. Huang, Q. Jin, and Y. Li, Five-loop anomalous dimensions of cubic scalar theory from operator product expansion, arXiv:2508.13620.
  17. J. A. Gracey, A. Maier, P. Marquard, and Y. Schröder, Anomalous dimensions and critical exponents for the Gross-Neveu-Yukawa model at five loops, Phys. Rev. D 112, 085029 (2025).
  18. O. Schnetz, ϕ3 theory at six loops, Phys. Rev. D 112, 016028 (2025).
  19. N. N. Bogoliubov and O. S. Parasiuk, On the multiplication of the causal function in the quantum theory of fields, Acta Math. 97, 227 (1957).
  20. K. Hepp, Proof of the Bogolyubov-Parasiuk theorem on renormalization, Commun. Math. Phys. 2, 301 (1966).
  21. W. Zimmermann, Convergence of Bogolyubov’s method of renormalization in momentum space, Commun. Math. Phys. 15, 208 (1969).
  22. J. Koschinski, M. V. Polyakov, and A. A. Vladimirov, Leading infrared logarithms from unitarity, analyticity and crossing, Phys. Rev. D 82, 014014 (2010).
  23. S. Caron-Huot and M. Wilhelm, Renormalization group coefficients and the S-matrix, J. High Energy Phys. 12 (2016) 010.
  24. I. Z. Rothstein and M. Saavedra, Extracting the asymptotic behavior of S-matrix elements from their phases, J. High Energy Phys. 11 (2024) 155.
  25. J. Elias Miró, J. Ingoldby, and M. Riembau, EFT anomalous dimensions from the S-matrix, J. High Energy Phys. 09 (2020) 163.
  26. P. Baratella, C. Fernandez, and A. Pomarol, Renormalization of higher-dimensional operators from on-shell amplitudes, Nucl. Phys. B959, 115155 (2020).
  27. M. Jiang, T. Ma, and J. Shu, Renormalization group evolution from on-shell SMEFT, J. High Energy Phys. 01 (2021) 101.
  28. Z. Bern, J. Parra-Martinez, and E. Sawyer, Structure of two-loop SMEFT anomalous dimensions via on-shell methods, J. High Energy Phys. 10 (2020) 211.
  29. P. Baratella, S. Maggio, M. Stadlbauer, and T. Theil, Two-loop infrared renormalization with on-shell methods, Eur. Phys. J. C 83, 751 (2023).
  30. L. C. Bresciani, G. Levati, P. Mastrolia, and P. Paradisi, Anomalous dimensions via on-shell methods: Operator mixing and leading mass effects, Phys. Rev. D 110, 056041 (2024).
  31. S. De Angelis and G. Durieux, EFT matching from analyticity and unitarity, SciPost Phys. 16, 071 (2024).
  32. L. C. Bresciani, G. Brunello, G. Levati, P. Mastrolia, and P. Paradisi, Renormalization of effective field theories via on-shell methods: The case of axion-like particles, J. High Energy Phys. 10 (2025) 190.
  33. Z.-W. Mo, S.-A. Wang, and J.-H. Yu, One-loop anomalous dimensions of LEFT dimension-7 operators via on-shell methods, Commun. Theor. Phys. 77, 115202 (2025).
  34. J. Aebischer, L. C. Bresciani, and N. Selimovic, Anomalous dimension of a general effective gauge theory. Part I. Bosonic sector, J. High Energy Phys. 08 (2025) 209.
  35. C. Itzykson and J. B. Zuber, Quantum Field Theory, International Series in Pure and Applied Physics (McGraw-Hill, New York, 1980).
  36. R. Britto, C. Duhr, H. S. Hannesdottir, and S. Mizera, Cutting-edge tools for cutting edges, in Encyclopedia of Mathematical Physics (Elsevier, 2025), pp. 595–620, 10.1016/b978-0-323-95703-8.00097-5.
  37. C. Itzykson, H. Saleur, and J. B. Zuber, Conformal invariance of nonunitary two-dimensional models, Europhys. Lett. 2, 91 (1986).
  38. Y.-t. Huang, D. A. McGady, and C. Peng, One-loop renormalization and the S-matrix, Phys. Rev. D 87, 085028 (2013).
  39. C. Cheung and C.-H. Shen, Nonrenormalization theorems without supersymmetry, Phys. Rev. Lett. 115, 071601 (2015).
  40. P. Baratella, D. Haslehner, M. Ruhdorfer, J. Serra, and A. Weiler, RG of GR from on-shell amplitudes, J. High Energy Phys. 03 (2022) 156.
  41. N. Arkani-Hamed, A. Hillman, and S. Mizera, Feynman polytopes and the tropical geometry of UV and IR divergences, Phys. Rev. D 105, 125013 (2022).
  42. M. Chala, Constraints on anomalous dimensions from the positivity of the S matrix, Phys. Rev. D 108, 015031 (2023).
  43. M. Chala and X. Li, Positivity restrictions on the mixing of dimension-eight SMEFT operators, Phys. Rev. D 109, 065015 (2024).
  44. Y.-P. Liao, J. Roosmale Nepveu, and C.-H. Shen, Positivity in perturbative renormalization: An EFT a-theorem, arXiv:2505.02910.
  45. L. Freidel, J. Padua-Argüelles, S. Schander, and M. Schiffer, Non-perturbative S-matrix renormalization, arXiv:2509.00156.
  46. H. Kleinert and V. Schulte-Frohlinde, Critical Properties of ϕ4-Theories (World Scientific, Singapore, 2001).
  47. O. Steinmann, Über den Zusammenhang zwischen den Wightmanfunktionen und den retardierten Kommutatoren, Helv. Phys. Acta 33, 257 (1960).
  48. O. Steinmann, Wightman-Funktionen und retardierte Kommutatoren. II, Helv. Phys. Acta 33, 347 (1960).
  49. A. Chavda, D. McLoughlin, S. Mizera, and J. Staunton, The unitary architecture of renormalization, arXiv:2511.10613.
  50. S. Arnone, A. Gatti, T. R. Morris, and O. J. Rosten, Exact scheme independence at two loops, Phys. Rev. D 69, 065009 (2004).
  51. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B435, 59 (1995).
  52. J.-y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, The rapidity renormalization group, Phys. Rev. Lett. 108, 151601 (2012).
  53. I. Z. Rothstein and M. Saavedra, A systematic Lagrangian formulation for quantum and classical gravity at high energies, arXiv:2412.04428.
  54. M. Kruczenski, J. Penedones, and B. C. van Rees, Snowmass white paper: S-matrix bootstrap, arXiv:2203.02421.
  55. C. de Rham, S. Kundu, M. Reece, A. J. Tolley, and S.-Y. Zhou, Snowmass white paper: UV constraints on IR physics, in Snowmass 2021 (2022), 3, arXiv:2203.06805.

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Phys. Rev. Lett. 136, 091604– Published 4 March, 2026
  • Received 17 November 2025
  • Revised 23 January 2026
  • Accepted 5 February 2026
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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.

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