Baryon transition form factors at the pole
Electromagnetic resonance properties are uniquely defined at the pole and do not depend on the separation of the resonance from background or the decay channel. Photon-nucleon branching ratios are nowadays often quoted at the pole, and we generalize the considerations to the case of virtual photons. We derive and compare relations for nucleon to baryon transition form factors both for the Breit-Wigner and the pole positions. Using the MAID2007 and SAID SM08 partial wave analyses of pion electroproduction data, we compare the ${G}_{M}, {G}_{E}$, and ${G}_{C}$ form factors for the $\mathrm{\ensuremath{\Delta}}(1232)$ resonance excitation at the Breit-Wigner resonance and pole positions up to ${Q}^{2}=5\phantom{\rule{0.16em}{0ex}}{\mathrm{GeV}}^{2}$. We also explore the $E/M$ and $S/M$ ratios as functions of ${Q}^{2}$. For pole and residue extraction, we apply the Laurent + Pietarinen method.