Chiral Lagrangians with $\mathrm{\Delta }(1232)$ to one loop

We construct the Lorentz-invariant chiral Lagrangians up to the order $\mathcal{O}(p^4)$ by including $\mathrm{\Delta }(1232)$ as an explicit degree of freedom. A full one-loop investigation on processes involving $\mathrm{\Delta }(1232)$ can be performed with them. For the $\pi \mathrm{\Delta }\mathrm{\Delta }$ Lagrangian, one obtains 38 independent terms at the order $\mathcal{O}(p^3)$ and 318 independent terms at the order $\mathcal{O}(p^4)$. For the $\pi N\mathrm{\Delta }$ Lagrangian, we get 33 independent terms at the order $\mathcal{O}(p^3)$ and 218 independent terms at the order $\mathcal{O}(p^4)$. The heavy baryon projection is also briefly discussed.