understood to be mass eigenstate fields) can be obtained as $$c^{s}_{ZH\eta}=\frac{\mathbb{R}_1 \mathbb{C}^{dH}\Upsilon
+\mathbb{R}_1 \mathbb{C}^{Hd}\Upsilon}{2}
\label{eq:cs}$$ Here we remark that we divide a general VSS vertex into its antisymmetric and symmetric parts because they exhibit distinct features in physical processes. For example, the symmetric VSS vertex does not contribute when the involved vector boson is on shell. Therefore, only the antisymmetric $ZH\eta$ vertex is expected to contribute at tree level to decay processes $H\rightarrow Z\eta$ (or $\eta\rightarrow ZH$ if $\eta$ is heavy) where $Z$ is supposed to be on shell.

Results
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In principle the derivation of mass eigenstate $ZH\eta$ vertex with no expansion on the $\frac{v}{f}$ can be carried out manually[^5]. However, after obtaining $V,F$ and $\mathbb{M}_V^2$, the calculation of $R$ and the inverse matrices can become extremely cumbersome. Therefore we choose to compute the mass eigenstate $ZH\eta$ vertex to $\ord ((\frac{v}{f})^3)$, which makes the results easier to obtain and display. For brevity we define $\xi\equiv\frac{v}{f}$ in the following.

Let us first find the scalar kinetic matrix $V$ and vector-scalar transition matrix $F$ for the SLH. They are computed to be $$\begin{aligned}
V= & \begin{pmatrix}
1 & 0 & \frac{\sqrt{2}}{t_{2\beta}}\xi-\frac{7c_{2\beta}+c_{6\beta}}{6\sqrt{2}s_{2\beta}^3}\xi^3 & -\sqrt{2}\xi+\frac{5+3c_{4\beta}}{3\sqrt{2}s_{2\beta}^2}\xi^3 \\
0 & 1 & -\frac{1}{\sqrt{2}}\xi+\frac{5+3c_{4\beta}}{12\sqrt{2}s_{2\beta}^2}\xi^3 & -\frac{2\sqrt{2}}{3t_{2\beta}}\xi^3 