Improved empirical parametrizations of the $\gamma^\ast N \to N(1535)$ transition amplitudes and the Siegert's theorem

Some empirical parametrizations of the $\gamma^\ast N \to N(1535)$ transition amplitudes violates the Siegert's theorem, that relates the longitudinal and the transverse amplitudes, in the pseudo-threshold limit (nucleon and resonance at rest). In the case of the electromagnetic transition from the nucleon (mass $M$) to the resonance $N(1525)$ (mass $M_R$), the Siegert's theorem is sometimes expressed by the relation $|{\bf q}| A_{1/2}= \lambda S_{1/2}$ in the pseudo-threshold limit, when the photon momentum $|{\bf q}|$ vanishes, and $\lambda = \sqrt{2} (M_R -M)$. In this article, we argue that the Siegert's theorem should be expressed by the relation $A_{1/2} = \lambda \frac{S_{1/2}}{ |{\bf q}|}$, in the limit $|{\bf q}| \to 0$. This result is a consequence of the relation $S_{1/2} \propto |{\bf q}|$, when $|{\bf q}| \to 0$, as suggested by the analysis of the transition form factors and by the orthogonality between the nucleon and $N(1535)$ states. We propose then new empirical parametrizations for the $\gamma^\ast N \to N(1535)$ helicity amplitudes, that are consistent with the data and the Siegert's theorem. The proposed parametrization follow closely the MAID2007 parametrization, except for a small deviation in the amplitudes $A_{1/2}$ and $S_{1/2}$ when $Q^2 < 1.5$ GeV$^2$.