Dispersive analysis of low energy ${\gamma }^{\ast }N\to \pi N$ process

We use a dispersion representation based on unitarity and analyticity to study the low energy ${\gamma }^{\ast }N\to \pi N$ process in the $S_{11}$ channel. Final state interactions among the $\pi N$ system are critical to this analysis. The left-hand part of the partial wave amplitude is imported from $\mathcal{O}(p^2)$ chiral perturbation theory result. On the right-hand part, the final state interaction is calculated through Omn`es formula in $S$ wave. It is found that a good numerical fit can be achieved with only one subtraction parameter, and the eletroproduction experimental data of multipole amplitudes $E_{0+},S_{0+}$ in the energy region below $\mathrm{\Delta }(1232)$ are well described when the photon virtuality $Q^2\le 0.1{\mathrm{G}\mathrm{e}\mathrm{V}}^2$.