$3\times3$ matrix ${\boldsymbol{\Delta}}_{hHA}({p^{2}})$, $${\boldsymbol{\Delta}}_{hHA} = {\begin{pmatrix}}\Delta_{hh} &\Delta_{hH}    &\Delta_{hA}\\
            \Delta_{Hh} &\Delta_{HH}    &\Delta_{HA}\\
            \Delta_{Ah} &\Delta_{AH}    &\Delta_{AA} 
 {\end{pmatrix}}\label{eq:DeltaMat}.$$ The off-diagonal entries (for $i\neq j$) result in: $$\Delta_{ij}({p^{2}}) = \frac{{\hat{\Gamma}_}{ij}{\hat{\Gamma}_}{kk}- {\hat{\Gamma}_}{jk}{\hat{\Gamma}_}{ki}}{{\hat{\Gamma}_}{ii}{\hat{\Gamma}_}{jj}{\hat{\Gamma}_}{kk} +2{\hat{\Gamma}_}{ij}{\hat{\Gamma}_}{jk}{\hat{\Gamma}_}{ki}-{\hat{\Gamma}_}{ii}{\hat{\Gamma}_}{jk}^{2} -{\hat{\Gamma}_}{jj}{\hat{\Gamma}_}{ki}^{2}-{\hat{\Gamma}_}{kk}{\hat{\Gamma}_}{ij}^{2}} \label{eq:Dij}.$$ All 2-point vertex functions $\hat{\Gamma}{(p^{2})}$ depend on ${p^{2}}$ via Eq.(\[eq:IrepVert\]). Here we do not write the ${p^{2}}$-dependence explicitly for the purpose of a simpler notation, but the full ${p^{2}}$-dependence is implied also below. The solutions of the diagonal propagators, $\Delta_{ii}$, can be expressed in the following compact way: $$\begin{aligned}
 \Delta_{ii}{(p^{2})}&= \frac{{\hat{\Gamma}_}{jj}{\hat{\Gamma}_}{kk}-{\hat{\Gamma}_}{jk}^{2}}{-{\hat{\Gamma}_}{ii}{\hat{\Gamma}_}{jj}{\hat{\Gamma}_}{kk} + {\hat{\Gamma}_}{ii}{\hat{\Gamma}_}{jk}^{2} - 2{\hat{\Gamma}_}{ij}{\hat{\Gamma}_}{jk}{\hat{\Gamma}_}{ki} + {\hat{\Gamma}_}{jj}{\hat{\Gamma}_}{ki}^{2} +{\hat{\Gamma}_}{kk}{\hat{\Gamma}_}{ij}^{2}}\label{eq:Dii}\\
&= \frac{i}{{p^{2}}-m_i^{2} + {\hat{\Sigma}^{\rm{eff}}_{ii}}{(p^{2})}},\label{eq:Diieff}\end{aligned}$$ where the effective self-energy is introduced, $$\begin{aligned}
   {\hat{\Sigma}^{\rm{eff}}_{ii}}(p^{2}) &= \hat{\Sigma}_{ii}(p^{2}) - i \frac{2{\hat{\Gamma}_}{ij}(p^{2}){\hat{\Gamma}_}{jk}(p^{2}){\hat{\Gamma}_}{ki}(p^{2}) - {\hat{\Gamma}_}{ki}^{2}({p^{2}}){\hat{\Gamma}_}{jj}({p^{2}})-{\hat{\Gamma}_}{ij}^{2}({p^{2}}){\hat{\Gamma}_}{kk}({p^{2}})}{{\hat{\Gamma}_}{jj}({p^{2}}){\hat{\Gamma}_}{kk}({p^{2}})-{\hat{\Gamma}_}{jk}^{2}({p^{2}})} \label{eq:seff}.\end{aligned}$$ It contains the diagonal self-energy, ${\hat{\Sigma}_{ii}}$ (which exists already at 1-loop order), and the mixing 2-point functions (whose products only contribute to ${\hat{\Sigma}^{\rm{eff}}_{ii}}$ from 2-loop order onwards). Hence, replacing the pure self-energy $\hat{\Sigma}_{ii}$ by the effective one, ${\hat{\Sigma}^{\rm{eff}}_{ii}}$, includes also the $3\times 3$ mixing contributions to the diagonal propagator in Eq.(\[eq:Diieff\]) while preserving formally the structure of the propagator as in the unmixed case. In the limit of no mixing, the second term in Eq.(\[eq:seff\]) vanishes.

The composition of ${\hat{\Sigma}^{\rm{eff}}_{ii}}$ in terms of the unmixed and the mixing contributions in Eq.(\[eq:seff\]) can 