Critical Bubbles and Fluctuations at the Electroweak Phase Transition

We discuss the critical bubbles of the electroweak phase transition using an effective high-temperature 3-dimensional action for the Higgs field $\phi$. The separate integration of gauge and Goldstone boson degrees of freedom is conveniently described in the 't Hooft-Feynman covariant background gauge. The effective dimensionless gauge coupling $g_3(T{)}^2$ in the broken phase is well behaved throughout the phase transition. However, the behavior of the one-loop $Z(\phi )$ factors of the Higgs and gauge kinetic terms signalizes the breakdown of the derivative expansion and of the perturbative expansion for a range of small $\phi$ values increasing with the Higgs mass $m_H$. Taking a functional $S_z[\phi ]$ with constant $Z(\phi )=z$ instead of the full non-local effective action in some neighborhood of the saddlepoint we are calculating the critical bubbles for several temperatures. The fluctuation determinant is calculated to high accuracy using a variant of the heat kernel method. It gives a strong suppression of the transition rate compared to previous estimates.