the application of ${\textbf{U}}$ resembles the use of ${\hat{\textbf{Z}}}$-factors in Eq.(\[eq:Zextsum\]) for the purpose of implementing partial higher-order effects into an improved Born result. Yet, the rotation matrix ${\textbf{U}}$ introduced for effective couplings as a unitary approximation is conceptually different from the ${\hat{\textbf{Z}}}$-matrix arising from propagator corrections and introduced for the correct normalisation of the $S$-matrix.

Technical treatment of imaginary parts
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In order to account for complex momenta and imaginary parts of self-energies, we employ an expansion of the self-energies around the real part of the complex momentum, $$\begin{aligned}
 {p^{2}}&\equiv {p^{2}}_r + i {p^{2}}_i \label{eq:psqReIm},\\
 {\hat{\Sigma}_{ij}}({p^{2}}) &\simeq {\hat{\Sigma}_{ij}}({p^{2}}_r) + i {p^{2}}_i {\hat{\Sigma}^{'}}_{ij}({p^{2}}_r)\label{eq:SigijReIm},\end{aligned}$$ where ${\hat{\Sigma}^{'}}_{ij}{(p^{2})}\equiv \frac{d{\hat{\Sigma}}_{ij}{(p^{2})}}{d{p^{2}}}$. This expansion enables the calculation of self-energies at complex momentum in terms of self-energies evaluated at real momentum (e.g. by `FeynHiggs`). For the inclusion of all products of real and imaginary parts, we do not expand the effective self-energy from Eq.(\[eq:seff\]) directly according to Eq.(\[eq:psqReIm\]). Instead, in the same way as in Refs.[@KarinaPhD; @Fowler:2010eba; @Fuchs:2015jwa], we expand all $\hat{\Gamma}_{ij}{(p^{2})}$ in Eq.(\[eq:seff\]) individually before combining them into ${\hat{\Sigma}^{\rm{eff}}_{ii}}$.

Pole structure of propagator matrices {#sect:poles}
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In this section, we will first discuss in detail the solutions of the poles of the propagator matrix depending on 