Dispersive Analysis of Low Energy $\gamma N\to \pi N$ Process and Studies on the $N^{\ast }(890)$ Resonance

Abstract

We present a dispersive representation of the $\gamma N\to \pi N$ partial-wave amplitude based on unitarity and analyticity. In this representation, the right-hand-cut contribution responsible for $\pi N$ final-state-interaction effect are taken into account via an Omnés formalism with elastic $\pi N$ phase shifts as inputs, while the left-hand-cut contribution is estimated by invoking chiral perturbation theory. Numerical fits are performed in order to pin down the involved subtraction constants. It is found that good fit quality can be achieved with only one free parameter and the experimental data of the multipole amplitude $E_0^{+}$ in the energy region below the $\Delta (1232)$ are well described. Furthermore, we extend the $\gamma N\to \pi N$ partial-wave amplitude to the second Riemann sheet so as to extract the couplings of the $N^{\ast }(890)$. The modulus of the residue of the multipole amplitude $E_0^{+}$ ($S_{11pE}$) is $2.41\mathrm{mfm}\cdot \mathrm{Ge}{\mathrm{V}}^2$ and the partial width of $N^{\ast }(890)\to \gamma N$ at the pole is about $0.369\mathrm{MeV}$, which is almost the same as the one of $N^{\ast }(1535)$, indicating that $N^{\ast }(890)$ strongly couples to $\pi N$ system.