The gravitational waves from the first-order phase transition with a dimension-six operator

We investigate in details the gravitational wave (GW) from the first-order phase transition (PT) in the extended standard model of particle physics with a dimension-six operator, which is capable of exhibiting the recently discovered slow first-order PT in addition to the usually studied fast first-order PT. To simplify the discussion, it is sufficient to work with an example of a toy model with the sextic term, and we propose an unified description for both slow and fast first-order PTs. We next study the full one-loop effective potential of the model with fixed/running renormalization-group (RG) scales. Compared to the prediction of GW energy density spectrum from the fixed RG scale, we find that the presence of running RG scale could amplify the peak amplitude by amount of one order of magnitude while shift the peak frequency to the lower frequency regime, and the promising regime of detection within the sensitivity ranges of various space-based GW detectors shrinks down to a lower cut-off value of the sextic term rather than the previous expectation.