mass eigenstate $ZH\eta$ vertex in the SLH. With the current double exponential parametrization it is possible to demonstrate that $H$ does not mix with $\sigma$, and the scalar kinetic terms are already canonically-normalized in the CP-even sector. Also, the $\mu$ term gives $\eta$ a mass but does not introduce mass mixing between $\eta$ and other fields. According to our argument in the previous section this means that after all the diagonalization procedure is completed, the whole effect on $\eta$ is supposed to be a simple rescaling. This offers great convenience for the derivation of the mass eigenstate $ZH\eta$ vertex. The needed rescaling factor can be easily computed. Going back to the notation of Section \[sec:sys\], the inner product between two Goldstone bosons $G_i$ and $G_j$ in Eq.  satisfies $$\begin{aligned}
\langle G_i|G_j\rangle & =(U^{-1})_{ik}(U^{-1})_{jl}\langle S_k|S_l\rangle
=(U^{-1})_{ik}(U^{-1})_{jl}\delta_{kl}
=(U^{-1})_{ik}(U^{-1})_{jk} \nonumber \\
& =(V^{-1})_{ij}\end{aligned}$$ We employ the convention that $\eta,\zeta,\chi,\omega$ correspond to indices $1,2,3,4$ respectively, therefore $$\langle\eta|\eta\rangle=(V^{-1})_{11}$$ Consequently, the ultimate mass eigenstate field $\eta^m$ is related to $\eta$ through $$\eta=\sqrt{(V^{-1})_{11}}\eta^m
\label{eq:eta}$$

To obtain the mass eigenstate $ZH\eta$ vertex, we also need to know the component of $\eta^m$ in $\zeta,\chi,\omega$. For the case of the SLH, let us denote the CP-odd sector elements of the 