Unstable particles in finite volume: The broken phase of the $4$-d $O(4)$ non-linear $\sigma$-model

According to a proposal of L"uscher it is possible to determine elastic scattering phases in infinite volume from the energy spectrum of two-particle states in a periodic box. We demonstrate the applicability of this method in the broken phase of the 4-dimensional $O(4)$ non-linear $\sigma$-model in a Monte-Carlo study on finite lattices. This non-perturbative approach also permits the study of unstable particles, the $\text{sg}$-particle in our case. We observe the $\text{sg}$-resonance and extract its mass and its width.