Model-independent connections between pion photoproduction and Compton scattering in theΔ(1232)region

We exploit Grushin et al.'s independent determinations of the real and the imaginary parts of the pion photoproduction multipoles in the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ region, without the use of an approximate form of Watson's theorem, and obtain model-independent estimates for the Compton amplitudes for the resonance excitation off nucleons. We compare these with the results obtained from other analyses of photoproduction multipoles. We find the lower bound for the differential cross section using unitarity, and compare our results with available experiments, both the older-generation experiments, primarily at Bonn, and the preliminary results from the recent studies at the Saskatchewan Accelerator Lab. We explore consistency of photoproduction multipoles with the forward Compton scattering amplitude extracted from the measured total hadronic cross sections of photons in hydrogen, using the optical theorem. Finally, we discuss implications for future precision Compton studies of the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ excitation, in particular, attempts to measure the $E2\mathrm{}$ to $M1\mathrm{}$ amplitude ratio, in the nucleon-to-$\Delta$ electromagnetic transition, which will be feasible at new photon facilities such as the Brookhaven Laser Electron-Gamma Source, exploring the photon polarization observables. These amplitudes contain valuable information on the structure of nucleon and $\Delta$ baryons, of great topical interest. We vividly demonstrate here the inadequacy of the older generation of Compton scattering experiments, as their poor photon energy resolution and counting statistics limit the quality of physics extractable from the data. This urgently calls for newer-generation high-statistics experiments with high photon energy resolution, using the photon polarization as a powerful tool.