QCD corrections to the electroweak sphaleron rate

The electroweak sphaleron rate in the high temperature phase of the Standard Model is inversely proportional to the weak-isospin conductivity. So far, only electroweak interactions were included in its computation. Here we take into account quark scattering through strong interactions at leading-log order. These reduce the quark contribution to the conductivity by up to 15 %, and the total conductivity by up to 6 %.

💡 Research Summary

This paper revisits the calculation of the electroweak sphaleron rate Γₛₚₕ in the high‑temperature phase of the Standard Model, focusing on the weak‑isospin conductivity σ that appears inversely in the rate formula Γₚₕₛ ∝ 1/σ. Earlier works accounted only for electroweak (W‑boson) scattering when determining σ, even though roughly half of the weak‑isospin current is carried by quarks. Quarks also undergo strong (QCD) interactions, and the authors set out to quantify how these affect σ at leading‑log order.

The theoretical framework starts from the Vlasov‑Boltzmann equation, which describes the coupled dynamics of long‑wavelength gauge fields and hard (|p|∼T) particle distributions in a hot plasma. By integrating out hard modes one obtains the hard‑thermal‑loop (HTL) effective theory. In the non‑Abelian SU(2) case the collision term for the weak‑isospin current reduces, after projecting onto the l = 1 spherical harmonic sector, to a simple linear damping term characterized by a coefficient c_W. At leading‑log order c_W = γ + (g_W² T/2π) K, with γ determined implicitly by a logarithmic equation involving the weak Debye mass m_W.

The novel contribution of the paper is the inclusion of QCD scattering processes. At leading‑log order only t‑channel 2↔2 processes with soft gluon or quark exchange generate logarithmic enhancements ∝ ln(g_S⁻¹). The authors adopt the collision kernel derived in previous literature and also re‑derive it from a two‑particle‑irreducible (2PI) effective action (see Appendix C). The QCD collision term separates into two parts: (i) gluon‑exchange diagrams (quark–quark and quark–gluon scattering) and (ii) quark‑exchange diagrams (quark–gluon scattering and quark–antiquark annihilation). Both contributions are proportional to g_S² ln(g_S⁻¹) and involve the gluon Debye mass m_G and the asymptotic quark mass m_∞ as infrared regulators.

Assuming a diagonal SU(2) electric field, the authors write the left‑handed quark distribution as f = f_eq + f_eq(1 − f_eq) η, with η proportional to the electric field and an unknown function χ(p₀). Inserting the QCD collision kernels yields a second‑order differential equation for χ(p₀) (Eq. 3.13). This equation is solved using a variational method and numerical inversion, allowing the extraction of the corrected damping coefficient c_W → c_W + Δc_W. The numerical results show that the quark contribution to the conductivity σ_q is reduced by up to 15 % for realistic values of the strong coupling (g_S≈0.5–0.6), while the total weak‑isospin conductivity σ is lowered by up to 6 %. Consequently, the sphaleron rate, which scales as 1/σ, is increased by a comparable amount.

The paper discusses the implications of this correction for baryogenesis calculations. Since the sphaleron rate determines the washout of any pre‑existing baryon asymmetry around the electroweak crossover, a 6 % increase could affect the viable parameter space of many baryogenesis scenarios. The authors also note that higher‑order (next‑to‑leading‑log) corrections are expected to be subdominant, as the leading‑log contribution already captures the dominant infrared physics.

Appendices provide the numerical values of the couplings used, a derivation of the Vlasov‑Boltzmann equation from Schwinger‑Dyson equations, and the detailed 2PI derivation of the QCD collision term. In summary, the work delivers the first quantitative assessment of strong‑interaction effects on the electroweak sphaleron rate, refining the theoretical input needed for precise early‑Universe calculations.