Combined parametrization of the neutron electric form factor and the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors

Models based on $SU(6)$ symmetry breaking and large $N_c$ limit provide relations between the pion cloud contributions to the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors, electric ($G_E$) and Coulomb ($G_C$), and the neutron electric form factor $G_{En}$, suggesting that those form factors are dominated by the same physical processes. Those relations are improved in order to satisfy a fundamental constraint between the electric and Coulomb quadrupole form factors in the long wavelength limit, when the photon three-momentum vanishes (Siegert's theorem). Inspired by those relations, we study alternative parametrizations for the neutron electric form factor. The parameters of the new form are then determined by a combined fit to the $G_{En}$ and the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factor data. We obtain a very good description of the $G_E$ and $G_C$ data when we combine the pion cloud contributions with small valence quark contributions to the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors. The best description of the data is obtained when the second momentum of $G_{En}$ is $r_n^4 \simeq -0.4$ fm$^4$. We conclude that the square radii associated with $G_E$ and $G_C$, $r_E^2$ and $r_C^2$, respectively, are large, revealing the long extension of the pion cloud. We conclude also that those square radii are related by $r_E^2 - r_C^2 = 0.6 \pm 0.2$ fm$^2$. The last result is mainly the consequence of the pion cloud effects and Siegert's theorem.