be written in the alternative way [@KarinaPhD; @Fowler:2010eba; @Fuchs:2015jwa] $$\begin{aligned}
 {\hat{\Sigma}^{\rm{eff}}_{ii}}{(p^{2})}&= {\hat{\Sigma}_{ii}}{(p^{2})}+ \frac{\Delta_{ij}{(p^{2})}}{\Delta_{ii}{(p^{2})}}{\hat{\Sigma}_{ij}}{(p^{2})}+ \frac{\Delta_{ik}{(p^{2})}}{\Delta_{ii}{(p^{2})}}{\hat{\Sigma}_{ik}}{(p^{2})}\label{eq:seffratio}\end{aligned}$$ for $i,j,k$ all different. Eq.(\[eq:seffratio\]) represents the sum of the diagonal and off-diagonal self-energies involving a Higgs boson $i$ where the off-diagonal contributions are weighted by the ratios of the respective off-diagonal and the diagonal propagators. This expression follows from Eq.(\[eq:seff\]) with the help of the equality $$\begin{aligned}
 \frac{\Delta_{ij}}{\Delta_{ii}} &= -\frac{{\hat{\Gamma}_}{ij}{\hat{\Gamma}_}{kk}-{\hat{\Gamma}_}{jk}{\hat{\Gamma}_}{ki}}{{\hat{\Gamma}_}{jj}{\hat{\Gamma}_}{kk}-{\hat{\Gamma}_}{jk}^{2}}
 = -\frac{{\hat{\Sigma}_{ij}}\,(D_k+{\hat{\Sigma}_{kk}})-{\hat{\Sigma}_{jk}}{\hat{\Sigma}_{ki}}}{(D_j+{\hat{\Sigma}_{jj}})\,(D_k+{\hat{\Sigma}_{kk}})-{\hat{\Sigma}_{jk}}^{2}}
 ,\label{eq:ratioijii}\end{aligned}$$ where the shorthand $$\begin{aligned}
 D_i{(p^{2})}&= {p^{2}}- m_i^{2} \label{eq:Di}\end{aligned}$$ has been used, analogously for $j\leftrightarrow k$, and ${\hat{\Sigma}}_{ij}=-i{\hat{\Gamma}_}{ij}$ from Eq.(\[eq:IrepVert\]) with $i\neq j$ (for the special case of $(2\times 2)$ mixing, this relation can be directly read off from Eqs.(\[eq:Dii2x2\]-\[eq:seff2x2\]) given below).

On-shell wave function normalisation factors {#sect:onshellZ}
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### Complex poles

The complex poles ${\mathcal{M}^{2}}_{a}$, where $a = 1, 2, 3$, are determined as solutions of the equations $$\begin{aligned}
{p^{2}}-m_i^{2} + {\hat{\Sigma}^{\rm{eff}}_{ii}}{(p^{2})}= 0 ,\label{eq:complexpole}\end{aligned}$$ with $i = h, H, A$, see Sect. \[sect:poles\] below. While the propagators have poles for ${p^{2}}={\mathcal{M}^{2}}_a$, it is interesting to note that the ratios of off-diagonal and diagonal propagators $$R_{ij}^{(a)} := \frac{\Delta_{ij}{(p^{2})}}{\Delta_{ii}{(p^{2})}}\bigg\rvert_{{p^{2}}={\mathcal{M}^{2}}_a},\label{eq:Rija}$$ stay finite at the complex poles ${\mathcal{M}^{2}}_a,\,~a=1,2,3$.

### Z-matrix for on-shell properties of external particles {#sect:Zmatrix}

Higgs bosons appearing as *external* particles in a process 