  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.
