for the treatment of loop functions of complex arguments). The evaluation at the complex poles leads to numerically more stable results than the evaluation at real ${p^{2}}=M_{h_a}^{2}$, as well as to the physically equivalent choices of index assignments. The calculation of the ${\hat{\textbf{Z}}}$-factors of MSSM Higgs bosons can be performed with the program `FeynHiggs`.

Use of Z-factors for external states {#sect:ExtH}
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The ${\hat{\textbf{Z}}}$-factors have been introduced for the correct normalisation of matrix elements with external (on-shell) Higgs bosons $h_a,~{p^{2}}={\mathcal{M}^{2}}_a$. It should be noted that ${\hat{\textbf{Z}}}$ does not provide a unitary transformation between the basis of lowest-order states and the basis of loop-corrected mass eigenstates. The fact that ${\hat{\textbf{Z}}}$ is a non-unitary matrix is related to the imaginary parts appearing in the propagators of unstable particles. Using the ${\hat{\textbf{Z}}}$-matrix, one can express the one-particle irreducible (1PI) vertex functions ${\hat{\Gamma}_}{h_a}$ involving a loop-corrected mass eigenstate $h_1,h_2,h_3$ as an external particle as a linear combination of the 1PI vertex functions of the lowest-order states, ${\hat{\Gamma}_}i$: $$\begin{aligned}
 {\hat{\Gamma}_}{h_a}&={\hat{\textbf{Z}}}_{ah}{\hat{\Gamma}_}h+{\hat{\textbf{Z}}}_{aH}{\hat{\Gamma}_}H+{\hat{\textbf{Z}}}_{aA}{\hat{\Gamma}_}A + ...\label{eq:Zextsum}\\
 &=\sqrt{{\hat{Z}}_a}\,\left({\hat{Z}}_{ah}{\hat{\Gamma}_}h+{\hat{Z}}_{aH}{\hat{\Gamma}_}H+{\hat{Z}}_{aA}{\hat{\Gamma}_}A\right) + \dots \label{eq:Vertexha},\end{aligned}$$ where the ellipsis refers to additional terms arising from the mixing with Goldstone and vector bosons, which are not described by the ${\hat{\textbf{Z}}}$-matrix. Thus, the overall normalisation factor $\sqrt{{\hat{Z}}_a}$ 