- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

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
  • Access by Xinjiang University

Hard cutoff and gauge theories

V. Branchina1,2,*, F. Contino3,4,†, R. Gandolfo1,2,‡, and A. Pernace1,2,§

  • *Contact author: branchina@ct.infn.it
  • †Contact author: f.contino@ssmeridionale.it
  • ‡Contact author: riccardo.gandolfo@ct.infn.it
  • §Contact author: arcangelo.pernace@ct.infn.it
PDF

Phys. Rev. D 114, 045010 – Published 13 August, 2026

DOI: https://doi.org/10.1103/sr3r-dqjd

Abstract

According to usual calculations, the use of a hard cutoff Λ in gauge theories leads to a violation of gauge invariance. This seems to generate a tension between gauge theories and the Wilsonian effective field theory paradigm, where Λ has the physical meaning of ultimate scale of the theory, the scale above which the latter has to be replaced by its UV completion. In the present work, considering the Euler-Heisenberg correction to the free Maxwell action, we present a way to introduce the Wilsonian hard UV cutoff Λ that preserves gauge invariance at the quantum level. For both scalar and fermionic QED, we recover the well-known Euler-Heisenberg result obtained within proper-time regularization, apart from terms that are generically cutoff suppressed. These terms, periodic in the inverse background field, might become relevant in regimes where the latter probes scales not much smaller than Λ. On the theoretical side, the methods developed in the present work represent a first step toward a new (closer in spirit to the Wegner-Houghton construction) realization of the Wilsonian renormalization group program in gauge theories.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (66)

  1. K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4, 3174 (1971).
  2. K. G. Wilson, Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior, Phys. Rev. B 4, 3184 (1971).
  3. K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28, 240 (1972).
  4. F. J. Wegner and A. Houghton, Renormalization group equation for critical phenomena, Phys. Rev. A 8, 401 (1973).
  5. K. G. Wilson and J. B. Kogut, The renormalization group and the epsilon expansion, Phys. Rep. 12, 75 (1974).
  6. K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773 (1975).
  7. J. C. Ward, An identity in quantum electrodynamics, Phys. Rev. 78, 182 (1950).
  8. Y. Takahashi, On the generalized Ward identity, Nuovo Cimento 6, 371 (1957).
  9. J. C. Taylor, Ward identities and charge renormalization of the Yang-Mills field, Nucl. Phys. B33, 436 (1971).
  10. A. A. Slavnov, Ward identities in gauge theories, Theor. Math. Phys. 10, 99 (1972).
  11. W. Pauli and F. Villars, On the invariant regularization in relativistic quantum theory, Rev. Mod. Phys. 21, 434 (1949).
  12. C. Branchina, V. Branchina, F. Contino, and N. Darvishi, Dimensional regularization, Wilsonian RG, and the naturalness and hierarchy problem, Phys. Rev. D 106, 065007 (2022).
  13. C. Branchina, V. Branchina, and F. Contino, Physical tuning and naturalness, Phys. Rev. D 107, 096012 (2023).
  14. M. Bonini, M. D’Attanasio, and G. Marchesini, Ward identities and Wilson renormalization group for QED, Nucl. Phys. B418, 81 (1994).
  15. M. Bonini, M. D’Attanasio, and G. Marchesini, Renormalization group flow for SU(2) Yang-Mills theory and gauge invariance, Nucl. Phys. B421, 429 (1994).
  16. M. Bonini, M. D’Attanasio, and G. Marchesini, BRS symmetry for Yang-Mills theory with exact renormalization group, Nucl. Phys. B437, 163 (1995).
  17. M. Bonini, M. D’Attanasio, and G. Marchesini, BRS symmetry from renormalization group flow, Phys. Lett. B 346, 87 (1995).
  18. M. D’Attanasio and T. R. Morris, Gauge invariance, the quantum action principle, and the renormalization group, Phys. Lett. B 378, 213 (1996).
  19. M. Bonini, G. Marchesini, and M. Simionato, Beta function and infrared renormalons in the exact Wilson renormalization group in Yang-Mills theory, Nucl. Phys. B483, 475 (1997).
  20. M. Simionato, Gauge consistent Wilson renormalization group: Abelian case, Int. J. Mod. Phys. A 15, 2121 (2000).
  21. M. Simionato, Gauge consistent Wilson renormalization group. 2: NonAbelian case, Int. J. Mod. Phys. A 15, 2153 (2000).
  22. T. R. Morris, A gauge invariant exact renormalization group. 1, Nucl. Phys. B573, 97 (2000).
  23. M. Simionato, On the consistency of the exact renormalization group approach applied to gauge theories in algebraic noncovariant gauges, Int. J. Mod. Phys. A 15, 4811 (2000).
  24. T. R. Morris, A gauge invariant exact renormalization group. 2, J. High Energy Phys. 12 (2000) 012.
  25. H. Gies and C. Wetterich, Renormalization flow of bound states, Phys. Rev. D 65, 065001 (2002).
  26. H. Gies, Running coupling in Yang-Mills theory: A flow equation study, Phys. Rev. D 66, 025006 (2002).
  27. S. Arnone, A. Gatti, and T. R. Morris, A proposal for a manifestly gauge invariant and universal calculus in Yang-Mills theory, Phys. Rev. D 67, 085003 (2003).
  28. H. Gies and J. Jaeckel, Renormalization flow of QED, Phys. Rev. Lett. 93, 110405 (2004).
  29. H. Gies and J. Jaeckel, Chiral phase structure of QCD with many flavors, Eur. Phys. J. C 46, 433 (2006).
  30. S. Arnone, T. R. Morris, and O. J. Rosten, Manifestly gauge invariant QED, J. High Energy Phys. 10 (2005) 115.
  31. J. M. Pawlowski, Aspects of the functional renormalisation group, Ann. Phys. (N.Y.) 322, 2831 (2007).
  32. T. R. Morris and O. J. Rosten, Manifestly gauge invariant QCD, J. Phys. A 39, 11657 (2006).
  33. S. Arnone, T. R. Morris, and O. J. Rosten, A generalised manifestly gauge invariant exact renormalisation group for SU(N) Yang-Mills, Eur. Phys. J. C 50, 467 (2007).
  34. B. J. Schaefer, J. M. Pawlowski, and J. Wambach, The phase structure of the Polyakov–Quark-Meson model, Phys. Rev. D 76, 074023 (2007).
  35. C. S. Fischer, A. Maas, and J. M. Pawlowski, On the infrared behavior of Landau gauge Yang-Mills theory, Ann. Phys. (Amsterdam) 324, 2408 (2009).
  36. J. Braun, H. Gies, and J. M. Pawlowski, Quark confinement from color confinement, Phys. Lett. B 684, 262 (2010).
  37. H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852, 287 (2012).
  38. W. j. Fu, J. M. Pawlowski, and F. Rennecke, QCD phase structure at finite temperature and density, Phys. Rev. D 101, 054032 (2020).
  39. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  40. C. Branchina, V. Branchina, F. Contino, and A. Pernace, Does the cosmological constant really indicate the existence of a dark dimension?, Int. J. Geom. Methods Mod. Phys. 22, 2450305 (2025),
  41. C. Branchina, V. Branchina, F. Contino, and A. Pernace, Dark dimension and the effective field theory limit, Int. J. Geom. Methods Mod. Phys. 22, 2450303 (2025).
  42. C. Branchina, V. Branchina, F. Contino, and A. Pernace, Path integral measure and the cosmological constant, Phys. Rev. D 111, 105018 (2025).
  43. C. Branchina, V. Branchina, F. Contino, and A. Pernace, Path integral measure and RG equations for gravity, Phys. Rev. D 111, 125021 (2025).
  44. C. Branchina, V. Branchina, F. Contino, R. Gandolfo, and A. Pernace, Diffeomorphism invariance of the effective gravitational action, Phys. Rev. D 112, 045002 (2025).
  45. C. Branchina, V. Branchina, F. Contino, R. Gandolfo, and A. Pernace, Gravity and the Higgs boson mass, arXiv:2507.13832.
  46. G. Giacometti, D. Rizzo, and D. Zappala, Universal content of the proper time flow in scalar and Yang-Mills theories, Phys. Rev. D 113, 045020 (2026).
  47. G. Giacometti, K. Kowalska, D. Rizzo, E. M. Sessolo, and D. Zappala, Quantum gravity contributions to the gauge and Yukawa couplings in proper time flow, Phys. Rev. D 114, 016020 (2026).
  48. W. Heisenberg and H. Euler, Consequences of Dirac’s theory of positrons, Z. Phys. 98, 714 (1936).
  49. V. Weisskopf, The electrodynamics of the vacuum based on the quantum theory of the electron, Kong. Dan. Vid. Sel. Mat. Fys. Med. 14N6, 1 (1936).
  50. J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  51. W. Dittrich, One loop effective potentials in QED, J. Phys. A 9, 1171 (1976).
  52. G. V. Dunne, Heisenberg-Euler effective Lagrangians: Basics and extensions, arXiv:hep-th/0406216.
  53. G. V. Dunne, The Heisenberg-Euler effective action: 75 years on, Int. J. Mod. Phys. A 27, 1260004 (2012).
  54. H. Gies and F. Karbstein, An addendum to the Heisenberg-Euler effective action beyond one loop, J. High Energy Phys. 03 (2017) 108.
  55. V. Branchina, F. Contino, R. Gandolfo, and A. Pernace (to be published).
  56. W. J. de Haas and P. M. van Alphen, The dependence of the susceptibility of diamagnetic metals upon the field, Proc. Acad. Sci. Amsterdam 33, 1106 (1930).
  57. D. Shoenberg, Magnetic Oscillations in Metals (Cambridge University Press, Cambridge, England, 1984).
  58. D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976).
  59. S. Janecek, E. Krotscheck, and Y. Zhang, Two-dimensional Bloch electrons in perpendicular magnetic fields: An exact calculation of the Hofstadter butterfly spectrum, Phys. Rev. B 87, 235429 (2013).
  60. C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, Hofstadter’s butterfly and the fractal quantum Hall effect in Moiré superlattices, Nature (London) 497, 598 (2013).
  61. B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, Massive Dirac Fermions and hofstadter butterfly in a van der Waals heterostructure, Science 340, 1427 (2013).
  62. M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realization of the hofstadter Hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett. 111, 185301 (2013).
  63. M. Bocarsly, M. Uzan, I. Roy, S. Grover, J. Xiao, Z. Dong, M. Labendik, A. Uri, M. E. Huber, Y. Myasoedov, K. Watanabe, T. Taniguchi, B. Yan, L. S. Levitov, and E. Zeldov, De Haas–van Alphen spectroscopy and magnetic breakdown in Moiré graphene, Science 383, 42 (2024).
  64. V. S. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math. 100, 191 (1998).
  65. M. Oleszczuk, A symmetry preserving cutoff regularization, Z. Phys. C 64, 533 (1994).
  66. S. B. Liao, Connection between momentum cutoff and operator cutoff regularizations, Phys. Rev. D 53, 2020 (1996).

Outline

Information

Phys. Rev. D 114, 045010– Published 13 August, 2026
  • Received 12 June 2026
  • Accepted 13 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