ming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nspw "fig:") ![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nnkw "fig:")

The weak interaction between the $\Sigma$, $\Lambda$ and $N$ baryons and the pseudoscalar $\pi$ and $K$ mesons is described by the phenomenological Lagrangians:

$$\begin{aligned}
\label{eq:weakl}
{\mathcal{L}}_{\Lambda N\pi}^w=&-iG_Fm_\pi^2\overline{\Psi}_N(A+B\gamma^5)
{\vec{\tau}}\cdot\vec{\pi}\Psi_\Lambda
\\\nonumber
{\mathcal{L}}_{\Sigma N\pi}^w=&
-iG_Fm_\pi^2\overline{\Psi}_N(\vec{A}_{\Sigma_i}+\vec{B}_{\Sigma_i}\gamma^5)
\cdot\vec{\pi}\Psi_{\Sigma_i}\,,
\\\nonumber
{\cal L}^{w}_{NN K} =&
-iG_Fm_\pi^2 \, \left[ \, \overline{\psi}_{N} \left( ^0_1 \right)
\,\,( C_{K}^{PV} + C_{K}^{PC} \gamma_5) \,\,(\phi^{K})^\dagger
\psi_{N} \right.
\\ \nonumber
 & \left. + \, \overline{\psi}_{N} \psi_{N}
\,\,( D_{K}^{PV}
+ D_{K}^{PC}
\gamma_5 ) \,\,(\phi^{K})^\dagger \,\,
\left( ^0_1 \right) \right] \ ,\end{aligned}$$

where $G_Fm_\pi^2=2.21\times10^{-7}$ is the weak Fermi coupling constant, $\gamma$ are the Dirac matrices and $\tau$ the Pauli matrices. The index $i$ appearing in the $\Sigma$ field refers to the different isospurion states for the $\Sigma$ hyperon: $$\Psi_{\Sigma\frac12}=
\left(\begin{array}{c}-\sqrt{\frac23}\Sigma_+\\\frac{1}{\sqrt3}\Sigma_0\end{array}\right)\,,
~~
\Psi_{\Sigma\frac32}=
\left(\begin{array}{c}0\\-\frac{1}{\sqrt3}\Sigma_+\\\sqrt{\frac23}\Sigma_0\\\Sigma_-\,
\end{array}\right)\,.$$ The PV and PC structures, $\vec{A}_{\Sigma_i}$ and $\vec{B}_{\Sigma_i}$ contain the corresponding weak coupling constants together with the isospin operators $\tau^a$ for $\frac12\to\frac12$ transitions and $T^a$ for $\frac12\to\frac32$ transitions. The weak couplings $A=1.05$, $B=-7.15$, $A_{\Sigma\frac12}=-0.59$, $A_{\Sigma\frac32}=2.00$, $B_{\Sigma\frac12}=-15.68$, and $B_{\Sigma\frac32}=-0.26