QCD corrections to the electroweak sphaleron rate

TL;DR

This work studies QCD corrections to the electroweak sphaleron rate in the high-temperature Standard Model by incorporating leading-log strong interactions into the weak-isospin conductivity $σ$. Starting from the Vlasov-Boltzmann framework for hard thermal loops and including a QCD collision term derived via a 2PI effective action, the authors derive a linearized Boltzmann equation for left-handed quarks with a new strong-scattering contribution. They solve for the momentum-dependent departure through a function $F(x)$ (with $x=p_0/T$) and extract a key parameter $\kappa$ that encodes the relative impact of QCD scatterings; they provide both a numerical solution and compact analytic approximations, finding that QCD corrections reduce the quark part of $σ$ by about $10$–$15\%$ and the total $σ$ by up to $6\%$ near the electroweak crossover. The corrected conductivity enters the hot sphaleron rate as $\Gamma_{\rm sph} \propto 1/σ$, enabling a readily updated estimate of baryon number violation in the early Universe. The work also offers a practical analytic formula for $κ(a)$, where $a$ measures the relative strength of EW and QCD collisions, facilitating quick implementation in phenomenological studies.

Abstract

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 %.

Figures (4)

• Figure 1: Feynman diagrams contributing to the QCD collision term at leading-log order. The logarithmic enhancement is the result of apparent IR divergencies in the $t$-channel contribution to $2\leftrightarrow2$ scattering processes with soft gluon or quark exchanges. Interference terms and $s$-channel contributions are IR safe and therefore only contribute at leading order.
• Figure 2: Solutions for $F(x)$ at $T/M _ Z =10$. The solid line shows the results from direct numercal integration of eq. \ref{['eq:F equation']}, which is indistinguishable from the variational result with $N = 10$ basis functions. The dash-dotted line shows the $1/a$ expansion in eqs. \ref{['eq:Fapprox']}, and the dashed line the result of the $N = 2$ variational ansatz.
• Figure 3: Quark contribution to the isospin conductivity with and without the strong collision term. The ratio $\kappa$ is defined in eq. \ref{['eq:kappa def']}. Solid lines show results from direct numercial integration of eq. \ref{['eq:F equation']}, which are identical to the variational results with $N = 10$ basis functions. Dash-dotted lines show results of the $1/a$ expansion given in eqs. \ref{['eq:Fapprox']} and \ref{['eq:1/a_kappa']}, and dotted lines show the analytic approximation eq. \ref{['eq:kapprox']}. The gauge couplings $g_S$ and $g_W$ are evaluated at the renormalization scale $\mu = \pi T$, see app. \ref{['sec:couplings']} for details.
• Figure 4: Leading logarithmic contributions to the quark selfenergy. The cut lines represent Wightman functions while uncut lines represent (anti-)time ordered propagators. The thick double lines on the left-hand side denote exact propagators, which can be expanded perturbatively to obtain the two diagrams on the right-hand side, thus yielding the relevant logarithmically enhanced contributions.