  The Coleman-Weinberg (CW) renormalization scheme for renormalization-group
improvement of the effective potential is particularly valuable for CW
symmetry-breaking mechanisms (including the challenging case of models with
multiple scalar fields). CW mechanism is typically studied using models with
classical scale invariance which not only provide a possibility for an
alternative symmetry breaking mechanism but also partially address the gauge
hierarchies through dimensional transmutation. As outlined in our discussion
section, when the couplings are not large, models with CW symmetry-breaking
mechanisms have also been shown to naturally provide the strong first-order
phase transition necessary for stochastic gravitational wave signals. A full
understanding of the CW-MS scheme transformation of couplings thus becomes
important in the era of gravitational wave detection and precision coupling
measurements. A generalized Coleman-Weinberg (GCW) renormalization scheme is
formulated and methods for transforming scalar self-couplings between the GCW
and MS (minimal-subtraction) renormalization schemes are developed. Scalar
$\lambda\Phi^4$ theory with global $O(4)$ symmetry is explicitly studied up to
six-loop order to explore the magnitude of this scheme transformation effect on
the couplings. The dynamical rescaling of renormalization scales between the
GCW and MS schemes can lead to significant (order of 10\%) differences in the
coupling at any order, and consequently GCW-MS scheme transformation effects
must be considered within precision determinations of scalar couplings in
extensions of the Standard Model.
