carry those two-point vector-scalar transitions, which are quite inconvenient. Nevertheless, the omission of $\Theta'$ and heavy sector gauge bosons can indeed be convenient if we only need to obtain the $\ord(\frac{v}{f})$ coefficient of the mass eigenstate $ZH\eta$ vertex, since the effect of those omitted two-point vector-scalar transitions will be suppressed due to the heavy gauge boson masses. Secondly, we have chosen to parameterize $\Phi_1,\Phi_2$ with two exponentials for each, rather than use a single exponential like $$\begin{aligned}
\Phi_{1,SE}=\exp\left[\frac{i}{f}(\Theta'+t_\beta\Theta)\right]
\begin{pmatrix}
0 \\ 0 \\ fc_\beta
\end{pmatrix}\end{aligned}$$ Also, in Eq.  and Eq.  the exponential of $\Theta'$ has been put to the left of the exponential of $\Theta$. For noncommutative matrices the single exponential parametrization is not mathematically equivalent to the double exponential parametrization. Moreover, the double exponential parametrization will depend on the order of the two exponentials. However, these parametrizations are related to each other by field redefinition and should thus be physically equivalent. Which one to use is a matter of convenience. We choose the double exponential parametrization in Eq.  and Eq.  because it does not introduce mass mixing between heavy and light sector scalars and will thus facilitate the mass diagonalization.

The aim of this section is to derive the 