$\pi N$ and $\pi \pi N$ Couplings of the $\mathrm{\Delta }(1232)$ and its Chiral Partners

We investigate the interactions and chiral properties of the four spin-$\text{thalf}$-baryons : $N^{-}(D_{13})$, $N^{+}(P_{13})$, ${\mathrm{\Delta }}^{+}(P_{33})$ and ${\mathrm{\Delta }}^{-}(D_{33})$ together with the nucleon. We construct the $SU(2)_R \times SU(2)_L$invariantinteractionsbetweenthespin-$\half$and-$\thalf$ baryons with the aid of a new, specially developed spin and isospin projection technique for these baryon fields, where the chiral invariant interactions contain one- and two-pion couplings. We obtain simple relations for the coupling constants of the one- and two-pion spin $\text{half}-\text{thalf}$ transitions terms. The relation for the one-pion interactions reasonably agrees with the experiments, which suggests that these spin-$\text{thalf}$ baryons are chiral partners.