Scale-genesis by Dark Matter and Its Gravitational Wave Signal

Classical scale invariance (CSI) may shed light on the weak scale origin, but the realistic CSI extension to the standard model requires a bosonic trigger. We propose a scalar Dark Matter(DM) field $X$ as the trigger, establishing a strong connection between the successful radiative breaking of CSI and DM phenomenologies. The latter forces the breaking scale to $\sim {\cal O}(\rm TeV)$. It brightens the test prospect of this scenario via gravitational wave, a sharp prediction of CSI phase transition (CSIPT), which is first order and has strong supercooling. Moreover, we carefully deal with some techniques which are commonly used to analyze CSIPT but maybe missed. In particular, we clarify the imprecision of Witten's formula used in the single field case to calculate the bubble nucleation rate and stress that the essence of Witten's approximation is the validity of high temperature expansion.