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

Konstantin T. Matchev1 and Sarunas Verner2

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Phys. Rev. D 112, 113009 – Published 30 December, 2025

DOI: https://doi.org/10.1103/nsrp-hwkg

Abstract

The electroweak sphaleron is a static, unstable solution of the Standard Model classical field equations, representing the energy barrier between topologically distinct vacua. In this work, we present a comprehensive updated analysis of the sphaleron using current Standard Model parameters with the physical Higgs boson mass of mH=125.1  GeV and mW=80.4  GeV, rather than the mH=mW approximation common in earlier studies. The study includes: (i) a complete derivation of the SU(2)×U(1) electroweak Lagrangian and field equations without gauge fixing constraints, (ii) high-precision numerical solutions for the static sphaleron configuration yielding a sphaleron energy Esph≃9.1  TeV, (iii) an analysis of the minimum energy path in field space connecting the sphaleron to the vacuum (a 1D “potential barrier” as a function of Chern-Simons number), and (iv) calculation of the sphaleron single unstable mode with negative eigenvalue ω−2=−2.7mW2, providing analytical fits for its eigenfunction. We find that using the measured Higgs mass modifies the unstable mode frequency, with important implications for baryon number violation rates in both early universe cosmology and potential high-energy collider signatures. These results provide essential input for accurate lattice simulations of sphaleron transitions and precision calculations of baryon number violation processes.

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Electroweak sphaleron revisited. II. Study of decay dynamics

Konstantin T. Matchev and Sarunas Verner
Phys. Rev. D 112, 113010 (2025)

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Phys. Rev. D 112, 113009– Published 30 December, 2025
  • Received 13 June 2025
  • Accepted 21 November 2025
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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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