Gauge-independent treatment of electroweak phase transition

We provide the first certificate of the gauge-independent bubble nucleation at the electroweak phase transition with the standard model effective field theory. Taking advantage of the thermal effective field theory framework, with the power counting $\lambda \sim g^3$, we rigorously demonstrate the gauge independence of the bubble nucleation rate up to two-loop order. Furthermore, we analyze the influence of relevant phase transition parameters on the gauge parameter and investigate its implications for gravitational waves generated by the electroweak phase transitions.