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 $\mathrm{\mid }\mathbf{q}\mathrm{\mid }{A}_{1\mathrm{/}2}=\lambda S_{1\mathrm{/}2}$ in the pseudo-threshold limit, when the photon momentum $\mathrm{\mid }\mathbf{q}\mathrm{\mid }$ vanishes, and $\lambda =\sqrt{2}(M_R-M)$. In this article, we argue that the Siegert’s theorem should be expressed by the relation $A_{1/2} = \lambda \frac{S_{1/2}}{ |{\bf q}|}$,inthelimit$|{\bf q}| \to 0$. This result is a consequence of the relation $S_{1\mathrm{/}2}\propto \mathrm{\mid }\mathbf{q}\mathrm{\mid }$, when $\mathrm{\mid }\mathbf{q}\mathrm{\mid }\to 0$, 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 $A_{1\mathrm{/}2}$ and $S_{1\mathrm{/}2}$ when $Q^2<1.5$ GeV${}^2$.