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

Abstract

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

Comments: To appear in the Phys.Lett.B. 7 pages, 4 figures