$\pi N$ $P_{11}$ wave amplitude in the Skyrme model

The quantized Skyrme model is applied to the $\pi N$ scattering problem, especially the $P_{11}$ wave amplitude. The field fluctuation and the zero mode are consistently taken into account, and the quantum correction to the Born term is calculated. It is found that this $1/N_c$ correction is comparable with the Born term and is crucially important for the $\pi N$ scattering problem in the Skyrme model.

Figure 1

$F\left(r\right)$ for the hedgehog configuration.

Figure 2

The radial function $R_{p0}^{\left(L\right)}$ for the dynamical pion (solid curve) at $p=296$ MeV/c (${\omega }_p=330$ MeV), and the spherical Bessel function $j_1\left(pr\right)$ (dashed curve).

Figure 3

The $\pi N$ $P_{11}$ amplitudes ${\mathcal{M}}_{\mathrm{Born}}$ (black solid curve), Re ${\mathcal{M}}_{Qc}^s$ (red [gray] solid curve), Im ${\mathcal{M}}_{Qc}^s$ (red [gray] dashed curve), and ${\mathcal{M}}_{Qc}^u$ (blue [gray] solid curve) are shown as functions of the pion energy ${\omega }_p$.