- 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

Electroweak sphaleron revisited. II. Study of decay dynamics

Konstantin T. Matchev1 and Sarunas Verner2

PDF

Phys. Rev. D 112, 113010 – Published 30 December, 2025

DOI: https://doi.org/10.1103/pxt7-hhlz

Abstract

We present a comprehensive analysis of electroweak sphaleron decay dynamics, employing both analytical techniques and high-resolution numerical simulations. Using a spherically symmetric ansatz, we reformulate the system as a (1+1)-dimensional problem and analyze its stability properties with current Standard Model parameters (mH=125.1  GeV, mW=80.4  GeV). We identify precisely one unstable mode with eigenvalue ω−2≃−2.7mW2 and numerically evolve the full nonlinear field equations under various initial conditions. Through spectral decomposition, we quantify the particle production resulting from the sphaleron decay. Our results demonstrate that the decay process is dominated by transverse gauge bosons, which constitute approximately 80% of the total energy and multiplicity, while Higgs bosons account for only 7%–8%. On average, the sphaleron decays into 49W bosons and 4 Higgs bosons. The particle spectra consistently peak at momenta k∼1–1.5mW, reflecting the characteristic size of the sphaleron. Remarkably, these properties remain robust across different decay scenarios, suggesting that the fundamental structure of the sphaleron, rather than specific triggering mechanisms, determines the decay outcomes. These findings provide distinctive experimental signatures of nonperturbative topological transitions in the electroweak theory, with significant implications for baryon number violation in the early universe and potentially for high-energy collider physics.

View figure in article

Physics Subject Headings (PhySH)

See Also

Electroweak sphaleron revisited. I. Static solutions, energy barrier, and unstable modes

Konstantin T. Matchev and Sarunas Verner
Phys. Rev. D 112, 113009 (2025)

Article Text

References (35)

  1. Stephen L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177, 2426 (1969).
  2. J. S. Bell and R. Jackiw, A PCAC puzzle: π0→γγ in the σ model, Nuovo Cimento A 60, 47 (1969).
  3. Frans R. Klinkhamer and Nicholas S. Manton, A saddle point solution in the Weinberg-Salam theory, Phys. Rev. D 30, 2212 (1984).
  4. Gerard ’t Hooft, Symmetry breaking through Bell-Jackiw anomalies, Phys. Rev. Lett. 37, 8 (1976).
  5. Gerard ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D 14, 3432 (1976); 18, 2199(E) (1978).
  6. V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, On the anomalous electroweak baryon number nonconservation in the Early Universe, Phys. Lett. 155B, 36 (1985).
  7. N. S. Manton, Topology in the Weinberg-Salam theory, Phys. Rev. D 28, 2019 (1983).
  8. Frans R. Klinkhamer and R. Laterveer, The sphaleron at finite mixing angle, Z. Phys. C 53, 247 (1992).
  9. Jutta Kunz, Burkhard Kleihaus, and Yves Brihaye, Sphalerons at finite mixing angle, Phys. Rev. D 46, 3587 (1992).
  10. Michela D’Onofrio, Kari Rummukainen, and Anders Tranberg, Sphaleron rate in the minimal standard model, Phys. Rev. Lett. 113, 141602 (2014).
  11. N. S. Manton and P. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2004).
  12. M. J. Gibbs, A. Ringwald, B. R. Webber, and J. T. Zadrozny, Monte Carlo simulation of baryon and lepton number violating processes at high-energies, Z. Phys. C 66, 285 (1995).
  13. A. Ringwald, Electroweak instantons/ sphalerons at VLHC?, Phys. Lett. B 555, 227 (2003).
  14. F. L. Bezrukov, D. Levkov, C. Rebbi, V. A. Rubakov, and P. Tinyakov, Semiclassical study of baryon and lepton number violation in high-energy electroweak collisions, Phys. Rev. D 68, 036005 (2003).
  15. F. L. Bezrukov, D. Levkov, C. Rebbi, V. A. Rubakov, and P. Tinyakov, Suppression of baryon number violation in electroweak collisions: Numerical results, Phys. Lett. B 574, 75 (2003).
  16. S. H. Henry Tye and Sam S. C. Wong, Bloch wave function for the periodic sphaleron potential and unsuppressed baryon and lepton number violating processes, Phys. Rev. D 92, 045005 (2015).
  17. John Ellis, Kazuki Sakurai, and Michael Spannowsky, Search for sphalerons: IceCube vs LHC, J. High Energy Phys. 05 (2016) 085.
  18. John Ellis and Kazuki Sakurai, Search for sphalerons in proton-proton collisions, J. High Energy Phys. 04 (2016) 086.
  19. Albert M Sirunyan et al. (CMS Collaboration), Search for black holes and sphalerons in high-multiplicity final states in proton-proton collisions at s=13  TeV, J. High Energy Phys. 11 (2018) 042.
  20. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  21. M. Fukugita and T. Yanagida, Baryogenesis without grand unification, Phys. Lett. B 174, 45 (1986).
  22. Sacha Davidson, Enrico Nardi, and Yosef Nir, Leptogenesis, Phys. Rep. 466, 105 (2008).
  23. Andrew G. Cohen, D. B. Kaplan, and A. E. Nelson, Progress in electroweak baryogenesis, Annu. Rev. Nucl. Part. Sci. 43, 27 (1993).
  24. V. A. Rubakov and M. E. Shaposhnikov, Electroweak baryon number nonconservation in the early universe and in high-energy collisions, Usp. Fiz. Nauk 166, 493 (1996).
  25. Mark Trodden, Electroweak baryogenesis, Rev. Mod. Phys. 71, 1463 (1999).
  26. David E. Morrissey and Michael J. Ramsey-Musolf, Electroweak baryogenesis, New J. Phys. 14, 125003 (2012).
  27. M. E. Shaposhnikov, Baryon asymmetry of the universe in standard electroweak theory, Nucl. Phys. B287, 757 (1987).
  28. M. Hellmund and J. Kripfganz, The decay of the sphalerons, Nucl. Phys. B373, 749 (1992).
  29. Konstantin T. Matchev and Sarunas Verner, preceding paper, Electroweak sphaleron revisited. I. Static solutions, energy barrier, and unstable modes, Phys. Rev. D 112, 113009 (2025).
  30. Margaret E. R. James, The sphaleron at nonzero Weinberg angle, Z. Phys. C 55, 515 (1992).
  31. D. Comelli, D. Grasso, M. Pietroni, and A. Riotto, The sphaleron in a magnetic field and electroweak baryogenesis, Phys. Lett. B 458, 304 (1999).
  32. Bharat Ratra and Laurence G. Yaffe, Spherically symmetric classical solutions in SU(2) gauge theory with a Higgs field, Phys. Lett. B 205, 57 (1988).
  33. T. Akiba, H. Kikuchi, and T. Yanagida, Static minimum energy path from a vacuum to a sphaleron in the Weinberg-Salam model, Phys. Rev. D 38, 1937 (1988).
  34. T. Akiba, H. Kikuchi, and T. Yanagida, The free energy of the sphaleron in the Weinberg-Salam model, Phys. Rev. D 40, 588 (1989).
  35. Yi-Zen Chu, James B. Dent, and Tanmay Vachaspati, Magnetic helicity in sphaleron debris, Phys. Rev. D 83, 123530 (2011).

Outline

Information

Phys. Rev. D 112, 113010– Published 30 December, 2025
  • Received 13 June 2025
  • Accepted 21 November 2025
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