R^T)_{pq}=\delta_{pq},\forall p,q=1,2,...,n_M\end{aligned}$$

We could further extend the definition of $G_p^m$ to the states $\bar{S}_r,r=n_M+1,...,n_S$ which we have already obtained. According to our diagonalization of physical scalar mass term, $\bar{S}_r$ can be expressed as $$\bar{S}_r=(PU)_{ri}G_i,r=n_M+1,...,n_S$$ where the matrix $U$ and $P$ are introduced in Eq.  and Eq. , respectively. Finally we can express $G_i^m$ as follows $$G_i^m=Q_{ij}G_j,i=1,2,...,n_S
\label{eq:Q1}$$ where the $n_S\times n_S$ matrix $Q$ is defined by (no summation over $i$) $$\begin{aligned}
Q_{ij}=\begin{cases}
\frac{(RF)_{ij}}{\mu_i}, & i=1,2,...,n_M, \\
(PU)_{ij}, & i=n_M+1,...,n_S.
\end{cases}
\label{eq:Q2}\end{aligned}$$ With the transformation matrix $R$ and $Q$ at our hand it will then be straightforward to derive any three-point or four-point interaction that we are interested in.

The case of SLH {#sec:slh}
===============

Preparation for the calculation
-------------------------------

The SLH model was proposed as a simple solution to the Higgs mass naturalness problem, making use of the collective symmetry breaking mechanism [@ArkaniHamed:2002qy]. Its electroweak gauge group is enlarged to $SU(3)_L\times U(1)_X$, and two scalar triplets are introduced to realize the global symmetry breaking pattern $$[SU(3)_1\times U(1)_1]\times[SU(3)_2\times U(1)_2]
\to[SU(2)_1\times U(1)_1]\times[SU(2)_2\times U(1)_2]$$ The scalar sector of the SLH model is usually written in a nonlinearly-realized form. In this paper we follow the convention of  [@delAguila:2011wk] and parameterize the two scalar triplets 