Δ(1232)electroproduction amplitudes in chiral soliton models of the nucleon

The multipole amplitudes for the $N-\Delta \mathrm{}\left(1232\right)\mathrm{}$ electromagnetic transition are computed in the framework of the linear $\sigma$ model and the chiral chromodielectric model for small and moderate photon virtualities. The models include quark and meson degrees of freedom and the nucleon and the $\Delta$ are clusters of three valence hedgehog quarks surrounded by meson clouds described by coherent states. Angular momentum and isospin projections are performed to endow model states representing the nucleon and the $\Delta$ with proper quantum numbers. Recoil corrections involved in the process ${\gamma }_{\mathrm{v}}\overrightarrow{N}\Delta$ are taken into account by performing linear momentum projection of the initial and final baryon states. The ratios $E2/M1\mathrm{}$ and $C2/M1\mathrm{}$ are in good agreement with the data in the two models, but the magnetic amplitude is better reproduced in the linear $\sigma$ model. The ratios show little dependence with the model parameters. Both in the linear $\sigma$ model and in the chromodielectric model the charged pions are responsible for the nonvanishing quadrupole-electric and -Coulomb amplitudes. The recoil corrections enhance the results obtained for the amplitudes without linear momentum projection, improving the comparison with experimental data. The dependence of the theoretical amplitudes with the choice of the reference frame is also studied.