so-called ‘$\mu$ term’ needs to be introduced $$\mL_\mu=\mu^2(\Phi_1^\dagger\Phi_2+\text{h.c.})$$ The observed Higgs-like boson will acquire its mass from the Coleman-Weinberg potential (however the $\mu$ term will also contribute to its potential). Because $\mL_{gk},\mL_\mu$ and the Coleman-Weinberg potential conserve CP, it will be convenient to group the neutral bosons into the CP-even and CP-odd sectors: $H,\sigma,Y_R$ belong to the CP-even sector, while $\eta,\zeta,\chi,\omega,Z',Z,Y_I,A$ belong to the CP-odd sector. There are no two-point transitions between these two sectors.

Some comments concerning the parametrization of $\Phi_1,\Phi_2$ in Eq.  and Eq.  are in order. Firstly, we have chosen to retain the heavy sector fields in $\Theta'$, rather than omitting them from the beginning. Apparently the omission of $\Theta'$ can be justified by doing a $SU(3)_L$ gauge transformation. This justification is valid, and in the more precise language of Faddeev-Popov gauge-fixing, the omission of $\Theta'$ actually corresponds to a certain choice of the gauge-fixing function. However, this omission could lead to future inconvenience, since as we will show, $\mL_{gk}$ contains two-point transitions between heavy sector gauge bosons and the pseudo-axion. $\Theta'$ can be rotated away by a gauge transformation but heavy sector gauge bosons cannot. This means that when doing perturbation theory we need to always 