parametrization and there are ‘unexpected’ vector-scalar two-point transitions that need to be taken care of. We formulated the problem in a generic setting as the diagonalization of a vector-scalar system in gauge field theories. Especially we proved that the scalar mass terms coming from the $R_\xi$ gauge-fixing procedure will be automatically orthogonal to each other if the corresponding gauge fields are rotated to their mass eigenstate prior to gauge-fixing. This fact greatly simplifies the diagonalization procedure.

For the SLH model, we found that the double exponential parametrization of scalar triplets, as shown in Eq.  and Eq.  is convenient for the derivation of $ZH\eta$ vertex, since in this parametrization the $\eta$ field is only subject to a simple rescaling in the diagonalization procedure, with which we could display in a simple form the $\eta^m$ component contained in the original $\eta,\zeta,\chi,\omega$ fields we started with, as shown in Eq. .

In principle the derivation of mass eigenstate $ZH\eta$ vertex could be worked out to all order in $\xi\equiv\frac{v}{f}$, however the intermediate results are too lengthy and we find it convenient to display the derivation and results to $\ord(\xi^3)$. The final results of antisymmetric and symmetric $ZH\eta$ vertices are shown in Eq.  and 