\label{eq:Zha}\,.\end{aligned}$$ Considering a diagram with the Higgs boson $i$ on an external line, whose propagator has three poles (see Sect. \[sect:poles\]), there are three possibilities which residue to compute. If the amputated Green’s function is evaluated at ${\mathcal{M}^{2}}_a$, the external $i$-line has to be multiplied by $\sqrt{{\hat{Z}}_i^{a}}$ for the correct S-matrix normalisation[^1]. So the resulting mass eigenstate as an outgoing particle is $h_a$. Alternatively, if the Green’s function is evaluated at ${\mathcal{M}^{2}}_b$, it has to be normalised by $$\begin{aligned}
\sqrt{{\hat{Z}}_i^{b}}=\frac{1}{\sqrt{1+{\hat{\Sigma}^{\rm{eff'}}}_{ii}({\mathcal{M}^{2}}_b)}}\end{aligned}$$ to achieve the correct S-matrix element. For this choice, the external mass eigenstate is $h_b$. Moreover, the wave function normalisation factor for $i-j$ mixing on an external on-shell line at ${\mathcal{M}^{2}}_a$ is composed of the overall normalisation factor $\sqrt{{\hat{Z}}_i^{a}}$ times the on-shell transition ratio $${\hat{Z}}_{ij}^{a} \equiv R_{ij}^{(a)} = \frac{\Delta_{ij}{(p^{2})}}{\Delta_{ii}{(p^{2})}}\bigg\rvert_{{p^{2}}={\mathcal{M}^{2}}_a} \label{eq:transij}\,.$$ Since $\Delta_{ii}$ and $\Delta_{ij}$ have – in the case of $3\times 3$ mixing – 3 complex poles (see Sect. \[sect:poles\]), any of them can be chosen for the evaluation of ${\hat{Z}}_i$, ${\hat{Z}}_{ij}$ and ${\hat{Z}}_{ik}$; in the considered example this is ${\mathcal{M}^{2}}_a$. Correspondingly, ${\hat{Z}}_j, {\hat{Z}}_{ji}$ and ${\hat{Z}}_{jk}$ will be evaluated at ${\mathcal{M}^{2}}_b$ and ${\hat{Z}}_k, {\hat{Z}}_{ki}$ and ${\hat{Z}}_{kj}$ at ${\mathcal{M}^{2}}_c$ where $a,b,c$ are a permutation of 1,2,3, and $i,j,k$ a permutation of 