mesons and baryons $$\label{phimatrix}
\Phi =
\left(
\begin{array}{ccc}
\frac{1}{\sqrt{2}} \pi^0 + \frac{1}{\sqrt{6}} \eta & \pi^+ & K^+ \\
\pi^- & - \frac{1}{\sqrt{2}} \pi^0 + \frac{1}{\sqrt{6}} \eta & K^0 \\
K^- & \bar{K}^0 & - \frac{2}{\sqrt{6}} \eta
\end{array}
\right)\, ,$$

$$\label{Bmatrix}
B =
\left(
\begin{array}{ccc}
\frac{1}{\sqrt{2}} \Sigma^0 + \frac{1}{\sqrt{6}} \Lambda &
\Sigma^+ & p \\
\Sigma^- & - \frac{1}{\sqrt{2}} \Sigma^0 + \frac{1}{\sqrt{6}} \Lambda & n \\
\Xi^- & \Xi^0 & - \frac{2}{\sqrt{6}} \Lambda
\end{array}
\right) \, .$$

The symbol $\langle \,\, \rangle$ stands for the $\textrm{SU(3)}$ trace and $f_\pi=93$ MeV is the pion decay constant. At energies close to threshold one can consider only the dominant contribution coming from $\partial_0$ and $\gamma^0$ [@OOR], such that the interaction is given by $$\begin{aligned}
\label{kernel}
V_{ij}= -C_{ij} \frac{1}{4f_\pi^2}(k^0+k^{\prime 0})\, ,\end{aligned}$$ where $k^0$, $k^{\prime 0}$ are the energies of the incoming and outgoing mesons, respectively, $$\label{k0}
  k^0 = \frac{s + m_{m_i}^2 -M_{B_i}^2}{2\sqrt{s}}\, , \quad \quad k^{\prime 0} = \frac{s + m_{m_j}^2 -M_{B_j}^2}{2\sqrt{s}}\, $$ where $m_{m_i}$, $M_{B_i}$ ($m_{m_j}$, $M_{B_j}$) are the masses of the initial (final) meson, baryon, respectively, and $C_{ij}$ are coefficients early calculated which are tabulated in Ref. [@mizutani] for the case of $K^- p$ and coupled channels. The extension of Eq.  to the 