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Nonperturbative determination of the sphaleron rate for first-order phase transitions

Jaakko Annala*, Kari Rummukainen†, and Tuomas V. I. Tenkanen‡

  • *Contact author: jaakko.annala@helsinki.fi
  • †Contact author: kari.rummukainen@helsinki.fi
  • ‡Contact author: tuomas.tenkanen@helsinki.fi
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Phys. Rev. D 113, 016014 – Published 13 January, 2026

DOI: https://doi.org/10.1103/q1jq-gq9m

Abstract

In many extensions of the Standard Model electroweak phase transitions at high temperatures can be described in a minimal dimensionally reduced effective theory with SU(2) gauge field and fundamental Higgs scalar. In this effective theory, all thermodynamic information is governed by two dimensionless ratios x≡λ3/g32 and y≡m32/g34, where λ3, m32, and g3 are the effective thermal scalar self-interaction coupling, the thermal mass and the effective gauge-coupling, respectively. By using nonperturbative lattice simulations to determine the rate of sphaleron transitions in the entire (x,y)-plane corresponding to the Higgs phase, and by applying previous lattice results for the bubble nucleation, we find a condition x(Tc)≲0.025 to guarantee preservation of the baryon asymmetry, which translates to v/Tc≡2Δ⟨ϕ†ϕ⟩/Tc≳1.33 for the (gauge-invariant) discontinuity in Higgs condensate. This indicates that viability of the electroweak baryogenesis requires the phase transition to be slightly stronger than previously anticipated. Finally, we present a general template for analysing such viability in a wide class of beyond the Standard Model theories, in which new fields are heavy enough to be integrated out at high temperature.

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Phys. Rev. D 113, 016014– Published 13 January, 2026
  • Received 8 September 2025
  • Accepted 22 December 2025
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