$\frac{E_{1+}}{M_{1+}}$ and $\frac{S_{1+}}{M_{1+}}$ and their $Q^2$ dependence in ${\gamma }_{v}N\to \Delta$ with relativized quark-model wave functions

We calculate the multipole ratios $\frac{E_{1+}}{M_{1+}}$ (electric quadrupole to magnetic dipole) and $\frac{S_{1+}}{M_{1+}}$ (scalar quadrupole to magnetic dipole) for the reaction ${\gamma }_{v}N\to \Delta$. We use the nonrelativistic transition operators but wave functions expanded in a large harmonic-oscillator basis which are the solutions of a "relativized" Hamiltonian (relativized wave functions), in order to test the dependence of the results on truncation of the wave-function basis. Our results show an $M_{1+}$ multipole closer to the experimental data at the photon point, $\frac{E_{1+}\mathrm{}\left(0\right)\mathrm{}}{M_{1+}\mathrm{}\left(0\right)\mathrm{}}\simeq -0.2\mathrm{}$, and smaller values of these ratios than previous calculations at all $Q^2$. The current-conservation condition between $S_{1+}$ and $L_{1+}$ (the longitudinal quadrupole multipole) is also satisfied to a better approximation with the relativized wave functions at values of $Q^2$ where the model is believable.