a lowest-order state ($i,j,k\in\{h,H,A\}$). Note that ${\hat{Z}}_{ai}={\hat{Z}}_{bj}={\hat{Z}}_{ck}\equiv1$ in the index scheme $I$ defined above. Once the index scheme has been specified, one can leave out the subscript $I$. For the scheme-independence of physical results, see Sect.\[sect:schemeIndep\].

Furthermore, it is convenient[@Frank:2006yh] to arrange the products of the normalisation factors $\sqrt{{\hat{Z}}_a}$ and transition ratios ${\hat{Z}}_{aj}$ as $$\begin{aligned}
{\hat{\textbf{Z}}}_{aj} &= \sqrt{{\hat{Z}}_a}{\hat{Z}}_{aj}
\label{eq:ZiZij}\end{aligned}$$ (note the difference between ${\hat{Z}}_{aj}$ and ${\hat{\textbf{Z}}}_{aj}$) into a non-unitary matrix: $$\begin{aligned}
 \hat{\textbf{Z}} &= {\begin{pmatrix}}\sqrt{\hat{Z}_1}{\hat{Z}}_{1h} & \sqrt{\hat{Z}_1}\hat{Z}_{1H} &\sqrt{\hat{Z}_1}\hat{Z}_{1A}\\
\sqrt{\hat{Z}_2}\hat{Z}_{2h} & \sqrt{\hat{Z}_2}{\hat{Z}}_{2H} &\sqrt{\hat{Z}_2}\hat{Z}_{2A}\\
\sqrt{\hat{Z}_3}\hat{Z}_{3h} & \sqrt{\hat{Z}_3}\hat{Z}_{3H} &\sqrt{\hat{Z}_3}{\hat{Z}}_{3A}
{\end{pmatrix}}\label{eq:Zmatrix}.\end{aligned}$$ The ${\hat{\textbf{Z}}}$-matrix defined in Eq.(\[eq:Zmatrix\]) fulfils the on-shell conditions (unit residue and vanishing mixing), which can be written in the following compact form[@Williams:2007dc; @KarinaPhD; @Fowler:2010eba; @Williams:2011bu]: $$\begin{aligned}
 \lim_{{p^{2}}\rightarrow {\mathcal{M}^{2}}_a} -\frac{i}{{p^{2}}-{\mathcal{M}^{2}}_a}\left({\hat{\textbf{Z}}}\cdot 
{\hat{\boldsymbol{\Gamma}}}_{hHA}\cdot {\hat{\textbf{Z}}}^{T} \right)_{hh} ~&= 1
\label{eq:Reshha},\\
 \lim_{{p^{2}}\rightarrow {\mathcal{M}^{2}}_b} -\frac{i}{{p^{2}}-{\mathcal{M}^{2}}_b}\left({\hat{\textbf{Z}}}\cdot 
{\hat{\boldsymbol{\Gamma}}}_{hHA}\cdot 
{\hat{\textbf{Z}}}^{T} \right)_{HH} &= 1
\label{eq:ResHHb},\\
 \lim_{{p^{2}}\rightarrow {\mathcal{M}^{2}}_c} -\frac{i}{{p^{2}}-{\mathcal{M}^{2}}_c}\left({\hat{\textbf{Z}}}\cdot 
{\hat{\boldsymbol{\Gamma}}}_{hHA}\cdot {\hat{\textbf{Z}}}^{T} \right)_{AA} &= 1
\label{eq:ResAAc},\end{aligned}$$ with vanishing off-diagonal entries. It is equally possible to begin with these equations (\[eq:Reshha\]-\[eq:ResAAc\]), i.e. to require unit residues and to demand vanishing mixing on-shell, to derive the elements of the ${\hat{\textbf{Z}}}$-matrix whose solutions are given in Eqs.(\[eq:Zha\]) and (\[eq:transij\]).

In Ref.[@Williams:2007dc] the evaluation at the full complex poles and the inclusion of imaginary parts were introduced (see Eq.(\[eq:SigijReIm\]) below 