systematic procedure of diagonalize a general vector-scalar system in gauge field theories will be elucidated. Then in Section \[sec:slh\] we apply this procedure to the SLH model and derive the mass eigenstate $ZH\eta$ vertex[^2] to $\ord\left((\frac{v}{f})^3\right)$. The $ZH\eta$ vertex derived here is found to be different from those which have already existed in the literature [@Kilian:2004pp; @Kilian:2006eh] for a long time. In Section \[sec:conc\] we present our discussion and conclusion.

General diagonalization procedure {#sec:sys}
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Consider a gauge field theory in which there are $n_S$ real scalar fields $G_i,i=1,2,...,n_S$ and $n_M$ real massive gauge boson fields $Z_p^{\mu},p=1,2,...,n_M$. If complex fields exist, we can always decompose them into their real components and proceed in a similar manner. The $G_i$’s which we start with neither need to be canonically normalized nor need to have diagonalized mass terms. For simplicity (but without loss of generality) the $Z_p$’s are assumed to have canonically normalized kinetic terms but don’t have to be diagonalized in their mass terms. When we say the $Z_p$’s are massive, it means that the eigenvalues of the mass matrix of $Z_p$’s are all positive. Especially, massless gauge bosons like photon are temporarily excluded from discussion. However, generalizing the procedure to theories 