Dynamical study of the $\Delta$ excitation in ${N(e,e}^{\prime }\pi )$ reactions

The dynamical model developed by us [Phys. Rev. C 54, 2660 (1996)] has been applied to investigate the pion electroproduction reactions on the nucleon. It is found that the model can describe to a very large extent the recent data of ${p(e,e}^{\prime }{\pi }^0)$ reaction from Jefferson Laboratory and MIT-Bates. The extracted magnetic dipole $\mathrm{}\left(M1\mathrm{}\right),$ electric dipole $\mathrm{}\left(E2\mathrm{}\right),$ and Coulomb $\mathrm{}\left(C2\mathrm{}\right)\mathrm{}$ strengths of the $\gamma \overrightarrow{N}\Delta$ transition are presented. It is found that the $C2/M1\mathrm{}$ ratio drops significantly with $Q^2$ and reaches about $-14\%$ at $Q^2=4\mathrm{}\hspace{0.5em}(\mathrm{GeV}{/c)}^2,$ while the $E2/M1\mathrm{}$ ratio remains close to the value $\sim -3\%$ at the $Q^2=0\mathrm{}$ photon point. The determined $M1\mathrm{}$ transition form factor drops faster than the usual dipole form factor of the proton. We also find that the nonresonant interactions can dress the $\gamma \overrightarrow{N}\Delta$ vertex to enhance strongly its strength at low $Q^2,$ but much less at high $Q^2.$ Predictions are presented for future experimental tests. Possible developments of the model are discussed.