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In the nucleon electroexcitation reactions, $\gamma^\ast N \to R$, where $R$is a nucleon resonance ($N^\ast$), the electric amplitude $E$, and thelongitudinal amplitude $S_{1/2}$, are related by $E \propto \frac{\omega}{|{\bfq}|}S_{1/2}$, at the pseudo-threshold limit ($|{\bf q}| \to 0$), where $\omega$and $|{\bf q}|$ are respectively the energy and the magnitude of three-momentumof the photon. The previous relation is usually refereed as the Siegert'stheorem. The form of the electric amplitude, defined in terms of the transverseamplitudes $A_{1/2}$ and $A_{3/2}$, and the explicit coefficients of therelation, depend on the angular momentum and parity ($J^P$) of the resonance$R$. The Siegert's theorem is the consequence of the structure of theelectromagnetic transition current, which induces constraints between theelectromagnetic form factors in the pseudo-threshold limit. In the presentwork, 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 betweenelectric amplitude and longitudinal amplitude, we obtain also a relationbetween the two transverse amplitudes: $A_{1/2}= A_{3/2} /\sqrt{3}$, at thepseudo-threshold. % The constraints at the pseudo-threshold are tested for theMAID2007 parametrizations of the reactions under discussion. Newparametrizations 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 areconsistent with both: the pseudo-threshold constraints (Siegert's theorem) andthe empirical data.

physical review. d/physical review. d.

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