Improved empirical parametrizations of the $\gamma^\ast N \to \Delta(1232)$ and $\gamma^\ast N \to N(1520)$ helicity amplitudes and the Siegert's theorem

In the nucleon electroexcitation reactions, $\gamma^\ast N \to R$, where $R$ is a nucleon resonance ($N^\ast$), the electric amplitude $E$, and the longitudinal amplitude $S_{1/2}$, are related by $E \propto \frac{\omega}{|{\bf q}|}S_{1/2}$, at the pseudo-threshold limit ($|{\bf q}| \to 0$), where $\omega$ and $|{\bf q}|$ are respectively the energy and the magnitude of three-momentum of the photon. The previous relation is usually refereed as the Siegert's theorem. The form of the electric amplitude, defined in terms of the transverse amplitudes $A_{1/2}$ and $A_{3/2}$, and the explicit coefficients of the relation, depend on the angular momentum and parity ($J^P$) of the resonance $R$. The Siegert's theorem is the consequence of the structure of the electromagnetic transition current, which induces constraints between the electromagnetic form factors in the pseudo-threshold limit. In the present work, we study the implications of the Siegert's theorem for the $\gamma^\ast N \to \Delta(1232)$ and $\gamma^\ast N \to N(1520)$ transitions. For the $\gamma^\ast N \to N(1520)$ transition, in addition to the relation between electric amplitude and longitudinal amplitude, we obtain also a relation between the two transverse amplitudes: $A_{1/2}= A_{3/2} /\sqrt{3}$, at the pseudo-threshold. % The constraints at the pseudo-threshold are tested for the MAID2007 parametrizations of the reactions under discussion. New parametrizations for the amplitudes $A_{1/2}$, $A_{3/2}$ and $S_{1/2}$, for the $\gamma^\ast N \to \Delta(1232)$ and $\gamma^\ast N \to N(1520)$ transitions, valid for small and large $Q^2$, are proposed. The new parametrizations are consistent with both: the pseudo-threshold constraints (Siegert's theorem) and the empirical data.