- 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
  • Open Access

Fixed points of the renormalization group running of quark and fermion mixing matrices in the standard model and beyond

Brian P. Dolan*

  • *Contact author: bdolan@stp.dias.ie
PDF

Phys. Rev. D 113, 095043 – Published 29 May, 2026

DOI: https://doi.org/10.1103/27bk-wwhd

Abstract

The renormalization group running of fermion mixing matrices in the Standard Model and beyond is studied. For the massless one-loop running with three generations, six fixed points are found. Their associated anomalous dimension matrices are calculated and the nature of each fixed point, whether attractive, repulsive, or mixed, is determined. An argument is given that the fixed points found at one loop must remain fixed points to all orders in perturbation theory and even nonperturbatively, as they are associated with certain differential geometric properties of vector fields on the space of mixing matrices. With Ng dark or sterile neutrinos there are at least Ng! fixed points of the fermion mixing matrix.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (35)

  1. H. D. Politzer, Reliable perturbative results for strong interactions, Phys. Rev. Lett. 30, 1346 (1973).
  2. D. J. Gross and F. Wilczek, Ultraviolet behavior of non-Abelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973).
  3. Y. Tang, Vacuum stability in the standard model, Mod. Phys. Lett. A 28, 1330002 (2013).
  4. F. Bezrukov and M. Shaposhnikov, Why should we care about the top quark Yukawa coupling?, J. Exp. Theor. Phys. 120, 335 (2015).
  5. M. Kobayashi and T. Maskawa, CP-violation in the renormalizable theory of weak interaction, Prog. Theor. Phys. 49, 652 (1973).
  6. B. Pontecorvo, Inverse beta processes and nonconservation of lepton charge, Zh. Eksp. Teor. Fiz. 34, 247 (1957) [Sov. Phys. JETP 7, 172 (1958)].
  7. Z. Maki, M. Nakagawa, and S. Sakata, Remarks on the unified model of elementary particles, Prog. Theor. Phys. 28, 870 (1962).
  8. E. Ma and S. Pakvasa, Cabibbo angle and quark masses in the Weinberg-Salam model, Phys. Lett. 86B, 43 (1979); Variation of mixing angles and masses with q2 in the standard six-quark model, Phys. Rev. D 20, 2899 (1979).
  9. B. Pendleton and G. G. Ross, Mass and mixing angle prediction from infra-red fixed points, Phys. Lett. 98B, 291 (1980).
  10. K. S. Babu, Renormalization-group analysis of the Kobayashi-Maskawa matrix, Z. Phys. C 35, 35 (1987).
  11. A. Denner and T. Sack, Renormalization of the quark mixing matrix, Nucl. Phys. B347, 203 (1990).
  12. C. Balzereit, T Hansmann, T. Mannel, and B. Plümper, The renormalization group evolution of the CKM matrix, Eur. Phys. J. C 9, 197 (1999).
  13. P. H. Chankowskia, W. Królikowski, and S. Pokorski, Fixed points in the evolution of neutrino mixings, Phys. Lett. B 473, 109 (2000).
  14. J. A. Casas, J. R. Espinosa, A. Ibarra, and I. Navarro, General RG equations for physical neutrino parameters and their phenomenological implications, Nucl. Phys. B573, 652 (2000).
  15. A. Pilaftsis, Gauge and scheme dependence of mixing matrix renormalization, Phys. Rev. D 65, 115013 (2002).
  16. R. N. Mohapatra, M. K. Parida, and G. Rajasekaran, High scale mixing unification and large neutrino mixing angles, Phys. Rev. D 69, 053007 (2004).
  17. D. Zhang, Threshold effects on the massless neutrino in the canonical seesaw mechanism, J. High Energy Phys. 10 (2024) 002.
  18. S. Antusch, J. Kersten, and M. Lindner and M. Ratz, Running neutrino masses, mixings and CP phases: Analytical results and phenomenological consequences, Nucl. Phys. B674, 401 (2003).
  19. Y. Grossman, A. Ismail, J. T. Ruderman, and T-H. Tsai, CKM substructure from the weak to the Planck scale, J. High Energy Phys. 06 (2022) 065.
  20. A. F. Pikelner and V. N. Velizhani, Yukawa coupling beta-functions in the standard model at three loops, Phys. Lett. B 722, 336 (2013).
  21. I. Jack and H. Osborn, Constraints on RG flow for four dimensional quantum field theories, Nucl. Phys. B883, 425 (2014).
  22. A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Three-loop SM beta-functions for matrix Yukawa couplings, Phys. Lett. B 737, 129 (2014).
  23. F. Herren, L. Mihaila, and M. Steinhauser, Gauge and Yukawa coupling beta functions of two-Higgs-doublet models to three-loop order, Phys. Rev. D 97, 015016 (2018); 101, 079903 (2020).
  24. D. Zhang, Two-loop renormalization group equations in the νSMEFT, J. High Energy Phys. 06 (2025) 106.
  25. R. Alkofer, A. Eichhorn, A. Held, C. M. Nieto, R. Percacci, and M. Schröfl, Quark masses and mixings in minimally parameterized UV completions of the standard model, Ann. Phys. (Amsterdam) 421, 168282 (2020).
  26. M. E. Machacek and M. T. Vaughn, Two-loop renormalization group equations in a general quantum field theory (II). Yukawa couplings, Nucl. Phys. B236, 221 (1984).
  27. M. Luo and Y. Yiao, Two-loop renormalization group equations in the standard model, Phys. Rev. Lett. 90, 011601 (2003).
  28. C. Jarlskog, Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation, Phys. Rev. Lett. 55, 1039 (1985).
  29. J. Schechter and J. W. F. Valle, Neutrino masses in SU(2)×U(1) theories, Phys. Rev. D 22, 2227 (1980).
  30. V. M. Buchstaber and S. Terzić, The foundations of (2n,k)- manifolds, Sbornik 210, 508 (2019).
  31. L. L. Chau and W.-Y. Keung, Comments on the parametrization of the Kobayashi-Maskawa matrix, Phys. Rev. Lett. 53 (1984) 1802.
  32. B. P. Dolan, Duality and the modular group in the quantum Hall effect, J. Phys. A 32, L243 (1999).
  33. S. Davidson, G. Isidori, and A. Strumi, The smallest neutrino mass, Phys. Lett. B 646, 100 (2007).
  34. S. Ray, W. Rodejohann, and M. A. Schmidt, Lower bounds on Ue3, Phys. Rev. D 83, 033002 (2011).
  35. D. J. Wallace and R. K. P. Zia, Gradient flow and the renormalisation group, Phys. Lett. 48A, 325 (1974); Gradient properties of the renormalisation group equations in multicomponent systems, Ann. Phys. (N.Y.) 92, 142 (1975).

Outline

Information

Phys. Rev. D 113, 095043– Published 29 May, 2026
  • Received 16 February 2026
  • Accepted 17 April 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