Improved empirical parametrizations of the ${\gamma }^{\ast }N\to \mathrm{\Delta }(1232)$ 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 $S_{1\mathrm{/}2}$, are related by $E \propto \frac{\omega}{|{\bf q}|}S_{1/2}$,atthepseudo-thresholdlimit($|{\bf q}| \to 0$),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 $A_{1\mathrm{/}2}$ and $A_{3\mathrm{/}2}$, 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\mathrm{.}Forthe$\gamma^\ast N \to N(1520)$ transition, in addition to the relation between electric amplitude and longitudinal amplitude, we obtain also a relation between the two transverse amplitudes: $A_{1\mathrm{/}2}=A_{3\mathrm{/}2}\mathrm{/}\sqrt{3}$, 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 $A_{1\mathrm{/}2}$, $A_{3\mathrm{/}2}$ and $S_{1\mathrm{/}2}$, for the ${\gamma }^{\ast }N\to \mathrm{\Delta }(1232)$ 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.