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

We construct the Lorentz-invariant chiral Lagrangians up to the order $\mathcal{O}(p^4)$ by including $\Delta(1232)$ as an explicit degree of freedom. A full one-loop investigation on processes involving $\Delta(1232)$ can be performed with them. For the $\pi\Delta\Delta$ 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 $\pi N\Delta$ 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.