need the appropriate on-shell properties for a correct normalisation of the S-matrix. In the hybrid on-shell/$\overline{\rm{DR}}$ renormalisation scheme [@Frank:2006yh], the masses are renormalised on-shell, but the $\overline{\textrm{DR}}$ renormalisation conditions for the fields and $\tan\beta$ [@Frank:2006yh], $$\begin{aligned}
 \delta Z_{\mathcal{H}_1}^{\overline{\rm{DR}}} &=& -\text{Re}\left[\Sigma^{'(\rm{div})}_{HH}(m_{H}^{2})\right]_{\alpha=0}\label{eq:dZH1},\\
 \delta Z_{\mathcal{H}_2}^{\overline{\rm{DR}}} &=& -\text{Re}\left[\Sigma^{'(\rm{div})}_{hh}(m_{h}^{2})\right]_{\alpha=0}\label{eq:dZH2},\\
 \delta \tan\beta^{\overline{\rm{DR}}} &=& \frac{1}{2}(\delta Z_{\mathcal{H}_2}^{\overline{\rm{DR}}} - \delta Z_{\mathcal{H}_1}^{\overline{\rm{DR}}})\label{eq:dtanb},\end{aligned}$$ where $\alpha$ is the mixing angle between $h$ and $H$, see Eq.(\[eq:ncHmix\]) below, do not ensure proper on-shell properties of the Higgs bosons. In fact, the loop-corrected mass eigenstates $h_1,h_2,h_3$, which occur as external, on-shell, particles e.g. in decay processes, are a mixture of the lowest order states $h,H,A$. Thus, finite wave function normalisation factors need to be introduced in order to guarantee that the mixing vanishes on-shell and that the propagators of the external particles have unit residue.

These so-called $Z$-factors for a neutral Higgs boson $i=h,H,A$ on an external line are obtained from the residue of the propagators at the complex pole ${\mathcal{M}^{2}}_a,\,a=1,2,3$,[@Dabelstein:1995js; @Heinemeyer:2001iy] $$\begin{aligned}
 {\hat{Z}}_i^{a} &:= \textrm{Res}_{{\mathcal{M}^{2}}_a}\left\lbrace\Delta_{ii}{(p^{2})}\right\rbrace   \label{eq:ZhaRes}.
 \end{aligned}$$ Expanding ${\hat{\Sigma}^{\rm{eff}}_{ii}}({p^{2}})$ around the complex pole ${p^{2}}={\mathcal{M}^{2}}_a$, one obtains the diagonal propagator $$\begin{aligned}
  \Delta_{ii}(p^{2}) &=\frac{i}{p^{2}-m_i^{2}+{\hat{\Sigma}^{\rm{eff}}_{ii}}(p^{2})}
=\frac{i}{p^{2}-{\mathcal{M}^{2}}_a}\cdot
\frac{1}{1+{\hat{\Sigma}^{\rm{eff'}}_{ii}}({\mathcal{M}^{2}}_a)+ \mathcal{O}({p^{2}}- {\mathcal{M}^{2}}_a)}
\label{eq:DiiApprdSeff}\,.\end{aligned}$$ Performing the limit ${p^{2}}\rightarrow {\mathcal{M}^{2}}_a$ yields the residue of Eq.(\[eq:ZhaRes\]), $$\begin{aligned}
 {\hat{Z}}_i^{a}
 &= \frac{1}{\frac{\partial}{\partial {p^{2}}}\frac{i}{\Delta_{ii}{(p^{2})}}}\bigg\rvert_{{p^{2}}={\mathcal{M}^{2}}_a}
 =\frac{1}{1+{\hat{\Sigma}^{\rm{eff'}}_{ii}}({\mathcal{M}^{2}}_a)} 