accounts for the particle $h_a$ appearing at an external line. In addition, the factors ${\hat{Z}}_{ai}$ given in Eqs.(\[eq:transij\]) and (\[eq:Zidentification\]) as ratios of propagators at ${p^{2}}={\mathcal{M}^{2}}_a$ describe the transition between the states $h_a$ and $i$. The transition factor ${\hat{Z}}_{ai}$ occurs in a diagram where $h_a$ is the external particle, but $i$ directly couples to the vertex. All possibilities for $i=h,H,A$ need to be included for each $h_a$, hence the sum arises. This is depicted in Fig.\[fig:Zext\] (cf. also Refs.[@Hahn:2007fq; @KarinaPhD]).

at (2.5,-0.6) [${p^{2}}={\mathcal{M}^{2}}_a$]{}; (1,0)–(4,0); at (2.5,0.6) [$\boldsymbol{h_a}$]{}; (4.3,0) circle (0.3); (5.6,1)–(4.6,0.1); (5.6,-1)–(4.6,-0.1); at (5.5,0) [$\hat\Gamma_{{\color{blue} h_a}}$]{}; at (7.2,0) [= $\sqrt{{\hat{Z}}_{{\color{blue} a}}}\Bigg($]{};

at (2.5,-0.8) [${\hat{Z}}_{{\color{blue}a}{\color{OliveGreen}h } }$]{}; (1,0)–(2.2,0); (2.8,0)–(4,0); at (1.6,0.6) [$\boldsymbol{h_a}$]{}; at (3.4,0.6) [$\boldsymbol{h}$]{}; (2.5,0) circle (0.3); (5,1)–(4,0)–(5,-1); at (5,0) [$\hat\Gamma_{{\color{OliveGreen} h}}$]{}; at (6,0) [+]{};

at (2.5,-0.8) [${\hat{Z}}_{{\color{blue}a}{\color{OliveGreen}H } }$]{}; (1,0)–(2.2,0); (2.8,0)–(4,0); at (1.6,0.6) [$\boldsymbol{h_a}$]{}; at (3.4,0.6) [$\boldsymbol{H}$]{}; (2.5,0) circle (0.3); (5,1)–(4,0)–(5,-1); at (5,0) [$\hat\Gamma_{{\color{OliveGreen} H}}$]{}; at (6,0) [+]{};

at (2.5,-0.8) [${\hat{Z}}_{{\color{blue}a}{\color{OliveGreen}A } }$]{}; (1,0)–(2.2,0); (2.8,0)–(4,0); at (1.6,0.6) [$\boldsymbol{h_a}$]{}; at (3.4,0.6) [$\boldsymbol{A}$]{}; (2.5,0) circle (0.3); (5,1)–(4,0)–(5,-1); at (5,0) [$\hat\Gamma_{{\color{OliveGreen} A}}$]{};

at (25,0) [$\Bigg)_{{p^{2}}={\mathcal{M}^{2}}_a}$]{}; at (26.4,0) [$+\dots$]{};

Conveniently, Eq.(\[eq:Vertexha\]) can be written in matrix form for all $h_1,h_2,h_3$ as $${\begin{pmatrix}}{\hat{\Gamma}_}{h_1}\\{\hat{\Gamma}_}{h_2}\\{\hat{\Gamma}_}{h_3}{\end{pmatrix}}= {\hat{\textbf{Z}}}\cdot {\begin{pmatrix}}{\hat{\Gamma}_}{h}\\{\hat{\Gamma}_}{H}\\{\hat{\Gamma}_}{A}{\end{pmatrix}}+\dots\label{eq:vertZ}~.$$ In this way, propagator corrections at external legs are 