Measurements of the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ transition form factor and the ratio ${\sigma }_n/{\sigma }_p$ from inelastic electron-proton and electron-deuteron scattering

Measurements of inclusive electron-scattering cross sections using hydrogen and deuterium targets in the region of the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ resonance are reported. A global fit to these new data and previous data in the resonance region is also reported for the proton. Transition form factors have been extracted from the proton cross sections for this experiment over the four-momentum transfer squared range $1.64<\mathrm{}{Q}^2<6.75\mathrm{}\hspace{0.5em}(\mathrm{GeV}{/c)}^2$ and from previous data over the range $2.41<\mathrm{}{Q}^2<9.82\mathrm{}\hspace{0.5em}(\mathrm{GeV}{/c)}^2.$ The results confirm previous reports that the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ transition form factor decreases more rapidly with $Q^2$ than expected from perturbative QCD. The ratio of ${\sigma }_n/{\sigma }_p$ in the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ resonance region has been extracted from the deuteron data for this experiment in the range $1.64<\mathrm{}{Q}^2<3.75\mathrm{}\hspace{0.5em}(\mathrm{G}\mathrm{e}\mathrm{V}/{c)}^2$ and for a previous experiment in the range $2.4<\mathrm{}{Q}^2<7.9\mathrm{}\hspace{0.5em}(\mathrm{GeV}{/c)}^2.$ A study has been made of the model dependence of these results. This ratio ${\sigma }_n/{\sigma }_p$ for $\Delta \mathrm{}\left(1232\right)\mathrm{}$ production is slightly less than unity, while ${\sigma }_n/{\sigma }_p$ for the nonresonant cross sections is approximately 0.5, which is consistent with deep inelastic scattering results.