First-order electroweak phase transition with a gauge-invariant approach

We study the electroweak phase transition with a three-dimensional standard model effective field theory at the two-loop level under a gauge-invariant approach. We observe that the phase transition parameters obtained with the gauge-invariant approach deviate from that obtained in the Landau gauge at around the percent level. We give the strongly first-order phase transition condition for electroweak baryogenesis and find that the predicted gravitational wave signals would be reduced for the new physics scale <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="normal">Λ</a:mi><a:mo>≲</a:mo><a:mn>590</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>GeV</a:mi></a:math>, and the parameter spaces of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="normal">Λ</d:mi><d:mo>≲</d:mo><d:mn>570</d:mn><d:mtext> </d:mtext><d:mtext> </d:mtext><d:mi>GeV</d:mi></d:math> can be probed by the space-based interferometers, such as LISA, TianQin, and Taiji.