Pion-proton scattering and isospin breaking in the $\Delta^0-\Delta^{++}$ system

We determine the mass and width of the $\Delta^{++}\ (\Delta^0)$ resonance from data on $\pi^+ p\ (\pi^- p)$ scattering both, in the pole of the $S$-matrix and conventional Breit-Wigner approaches to the scattering amplitude. We provide a simple formula that relates the two definitions for the parameters of the $\Delta$. Isospin symmetry breaking in the $\d0-\dm$ system depends on the definition of the resonant properties: we find $M_0-M_{++} = 0.40 \pm 0.57\ {\rm MeV},\ \Gamma_0 -\Gamma_{++} = 6.89 \pm 0.95\ {\rm MeV}$ in the pole approach while $\wt{M}_0-\wt{M}_{++} = 2.25 \pm 0.68\ {\rm MeV},\ \wt{\Gamma}_0 - \wt{\Gamma}_{++} = 8.45 \pm 1.11\ {\rm MeV}$ in the conventional approach.