First-order Electroweak phase transition with Gauge-invariant approach

We study the electroweak phase transition dynamics with a three-dimensional standard model effective field theory under a gauge-invariant approach. We observe that, at the two-loop level, the phase transition parameters obtained with the gauge invariant approach can at most deviate from that of the dimensional reduction method around the percent level. We further found that the predicted gravitational wave signals at the new physics scale $\mathrm{\Lambda }\gtrsim 590$ GeV are unreachable by the space-based interferometers, such as: LISA, TianQin, and Taiji.