- 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

Acoustic phonons in a magnetized vacuum? First-principle lattice results on the mass spectrum of the electroweak model in a strong magnetic field

M. N. Chernodub1,2, V. A. Goy3,4, and A. V. Molochkov1,3,5

PDF

Phys. Rev. D 113, 074508 – Published 14 April, 2026

DOI: https://doi.org/10.1103/b4cm-y63c

Abstract

We use numerical Monte Carlo simulations to determine the mass spectrum of the bosonic sector of the electroweak model in an external magnetic field of the electroweak-scale strength (1020  T) at zero temperature. It is known that as the magnetic field gets stronger, the electroweak vacuum undergoes two consecutive crossover-type transitions, passing from (i) the conventional symmetry-broken homogeneous phase to (ii) an intermediate inhomogeneous vortex phase characterized by a (superconducting) condensate of electrically charged W bosons and then to (iii) a homogeneous phase with a restored electroweak symmetry. We show that the spin component of the W boson aligned with the direction of the magnetic field is the lightest excitation in all three phases. Its mass continuously decreases in the low-field broken phase and becomes very small in the intermediate phase. We argue that this nearly massless excitation corresponds to a Goldstone acoustic phonon mode associated with vibrations of the lattice of electroweak vortices. In the high-field symmetry-restored phase, where the vortices disappear, the lightest W mass rises again. Neither Higgs nor Z boson masses vanish across all studied phases and crossover transitions.

View figure in article

Physics Subject Headings (PhySH)

See Also

Phase Structure of Electroweak Vacuum in a Strong Magnetic Field: The Lattice Results

M. N. Chernodub, V. A. Goy, and A. V. Molochkov
Phys. Rev. Lett. 130, 111802 (2023)

Article Text

References (45)

  1. E. A. Paschos, Electroweak Theory (Cambridge University Press, Cambridge, England, 2007), 10.1017/cbo9780511611049.
  2. Paul Langacker, Introduction to the standard model and electroweak physics, in Theoretical Advanced Study Institute in Elementary Particle Physics : The Dawn of the LHC Era (2010), pp. 3–48, .
  3. A. Pich, The Standard model of electroweak interactions, in 2004 European School of High-Energy Physics (2005), pp. 1–48, .
  4. D. A. Kirzhnits and A. D. Linde, Macroscopic consequences of the Weinberg model, Phys. Lett. 42B, 471 (1972).
  5. Steven Weinberg, Gauge and global symmetries at high temperature, Phys. Rev. D 9, 3357 (1974).
  6. Michela D’Onofrio and Kari Rummukainen, Standard model cross-over on the lattice, Phys. Rev. D 93, 025003 (2016).

  7. We use the natural units ℏ=c=kB=1 in the paper.

  8. Viatcheslav Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, Cambridge, England, 2005), 10.1017/cbo9780511790553.
  9. T. Vachaspati, Magnetic fields from cosmological phase transitions, Phys. Lett. B 265, 258 (1991).
  10. Dario Grasso and Hector R. Rubinstein, Magnetic fields in the early universe, Phys. Rep. 348, 163 (2001).
  11. Juan Maldacena, Comments on magnetic black holes, J. High Energy Phys. 04 (2021) 079.
  12. Diptimoy Ghosh, Arun Thalapillil, and Farman Ullah, Astrophysical hints for magnetic black holes, Phys. Rev. D 103, 023006 (2021).
  13. Romain Gervalle and Mikhail S. Volkov, Black holes with electroweak hair, Phys. Rev. Lett. 133, 171402 (2024).
  14. V. V. Skalozub, Vacuum structures in Weinberg-Salam theory, Sov. J. Nucl. Phys. 45, 1058 (1987).
  15. Jan Ambjørn and P. Olesen, Antiscreening of large magnetic fields by vector bosons, Phys. Lett. B 214, 565 (1988).
  16. J. Ambjørn and P. Olesen, A condensate solution of the electroweak theory which interpolates between the broken and the symmetric phase, Nucl. Phys. B330, 193 (1990).
  17. J. Ambjørn and P. Olesen, On electroweak magnetism, Nucl. Phys. B315, 606 (1989).
  18. J. Ambjørn and P. Olesen, Electroweak magnetism: Theory and applications, Int. J. Mod. Phys. A 05, 4525 (1990).
  19. Jan Ambjørn and P. Olesen, Electroweak magnetism, W condensation and antiscreening, in 4th Hellenic School on Elementary Particle Physics (1992), pp. 396–406, .
  20. M. N. Chernodub, Superconductivity of QCD vacuum in strong magnetic field, Phys. Rev. D 82, 085011 (2010).
  21. M. N. Chernodub, Jos Van Doorsselaere, and Henri Verschelde, Electromagnetically superconducting phase of vacuum in strong magnetic field: Structure of superconductor and superfluid vortex lattices in the ground state, Phys. Rev. D 85, 045002 (2012).
  22. M. N. Chernodub, Jos Van Doorsselaere, and Henri Verschelde, Magnetic-field-induced superconductivity and superfluidity of W and Z bosons: In tandem transport and kaleidoscopic vortex states, Phys. Rev. D 88, 065006 (2013).
  23. V. V. Skalozub, On restoration of spontaneously broken symmetry in magnetic field, Yad. Fiz. 28, 228 (1978) [Sov. J. Nucl. Phys. 23, 113 (1978)].
  24. M. N. Chernodub, V. A. Goy, and A. V. Molochkov, Phase structure of electroweak vacuum in a strong magnetic field: The lattice results, Phys. Rev. Lett. 130, 111802 (2023).
  25. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  26. P. G. De Gennes and J. Matricon, Collective modes of vortex lines in superconductors of the second kind, Rev. Mod. Phys. 36, 45 (1964).
  27. V. K. Tkachenko, Stability of vortex lattices, Sov. Phys. JETP 23, 1049 (1966).
  28. Baruch Rosenstein and Dingping Li, Ginzburg-Landau theory of type II superconductors in magnetic field, Rev. Mod. Phys. 82, 109 (2010).
  29. Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor, Codata recommended values of the fundamental physical constants: 2018, J. Phys. Chem. Ref. Data 50, 033105 (2021).
  30. For a detailed description of the numerical procedures used in the numerical first-principle lattice simulations implemented in the paper, see the Supplemental Material at 10.1103/PhysRevLett.130.111802.
  31. Axel Maas, Brout-Englert-Higgs physics: From foundations to phenomenology, Prog. Part. Nucl. Phys. 106, 132 (2019).
  32. Christof Gattringer and Christian B. Lang, Quantum Chromodynamics on the Lattice (Springer, Berlin, 2010), Vol. 788.
  33. Stephan Durr, Gauge action improvement and smearing, Comput. Phys. Commun. 172, 163 (2005).
  34. V. V. Skalozub, Effective mass of W boson in a magnetic field, Phys. At. Nucl. 77, 901 (2014).
  35. Charles Kittel, Physical theory of ferromagnetic domains, Rev. Mod. Phys. 21, 541 (1949).
  36. A. Prabhakar and D. D. Stancil, Spin Waves: Theory and Applications (Springer-Verlag, New York, NY, 2008), 10.1007/978-0-387-77865-5.
  37. E H Brandt, The flux-line lattice in superconductors, Rep. Prog. Phys. 58, 1465 (1995).
  38. M. N. Chernodub, Jos Van Doorsselaere, and Henri Verschelde, Spontaneous electromagnetic superconductivity and superfluidity of QCDxQED vacuum in strong magnetic field, Proc. Sci. QNP2012 (2012) 109 [arXiv:1206.2845].
  39. N. K. Nielsen and P. Olesen, An unstable Yang-Mills field mode, Nucl. Phys. B144, 376 (1978).
  40. Jan Ambjørn and P. Olesen, A magnetic condensate solution of the classical electroweak theory, Phys. Lett. B 218, 67 (1989); 220, 659(E) (1989).
  41. Samuel W. MacDowell and Ola Tornkvist, Structure of the ground state of the electroweak gauge theory in a strong magnetic field, Phys. Rev. D 45, 3833 (1992).
  42. Abdus Salam and J. A. Strathdee, Transition electromagnetic fields in particle physics, Nucl. Phys. B90, 203 (1975).
  43. Andrei D. Linde, Symmetry behavior in external fields, Phys. Lett. 62B, 435 (1976).
  44. See information about the Shared Resource Center at https://cc.dvo.ru.
  45. See information about the Resources at http://ckp.nrcki.ru.

Outline

Information

Phys. Rev. D 113, 074508– Published 14 April, 2026
  • Received 28 January 2026
  • Accepted 24 March 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