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

TL;DR

This paper develops Lorentz-invariant chiral Lagrangians including the $ ext{Delta}(1232)$ resonance up to fourth order, enabling comprehensive one-loop calculations for processes involving this resonance.

Contribution

It constructs detailed chiral Lagrangians with explicit $ ext{Delta}(1232)$ degrees of freedom up to $ ext{O}(p^4)$, providing a foundation for advanced loop computations.

Findings

38 independent terms at $ ext{O}(p^3)$ for $ ext{pi} ext{Delta} ext{Delta}$

318 independent terms at $ ext{O}(p^4)$ for $ ext{pi} ext{Delta} ext{Delta}$

33 independent terms at $ ext{O}(p^3)$ for $ ext{pi} ext{N} ext{Delta}

Abstract

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.