Meson-exchange Model for $πN$ scattering and $γN -> πN$ reaction

An effective Hamiltonian consisting of bare $\Delta \leftrightarrow\pi N$, $\gamma N$ vertex interactions and energy-independent meson-exchange $\pi N \leftrightarrow \pi N, \gamma N$ transition operators is derived by applying a unitary transformation to a model Lagrangian with $N,\Delta,\pi$, $\rho$, $\omega$, and $\gamma$ fields. With appropraite phenomenological form factors and coupling constants for $\rho$ and $\Delta$, the model can give a good description of $\pi N$ scattering phase shifts up to the $\Delta$ excitation energy region. It is shown that the best reproduction of the recent LEGS data of the photon-asymmetry ratios in $\gamma p \rightarrow \pi ^0 p$ reactions provides rather restricted constraints on the coupling strengths $G_E$ of the electric $E2$ and $G_M$ of the magnetic $M1$ transitions of the bare $\Delta \leftrightarrow \gamma N$ vertex and the less well-determined coupling constant $g_{\omega NN}$ of $\omega$ meson. Within the ranges that $G_M = 1.9 \pm 0.05$, $G_E = 0.0 \pm 0.025$, and $7 \leq g_{\omega NN}\leq 10.5$, the predicted differential cross sections and photon-asymmetry ratios are in an overall good agreement with the data of $\gamma p \rightarrow \pi ^0 p$, $\gamma p \rightarrow \pi ^+ n$, and $\gamma n\rightarrow \pi ^- p$ reactions from 180 MeV to the $\Delta$ excitation region. The predicted $M_{1^+}$ and $E_{1^+}$ multipole amplitudes are also in good agreement with the empirical values determined by the amplitude analyses. The constructed effective Hamiltonian is free of the nucleon renormlization problem and hence is suitable for nuclear many-body calculations. We have also shown that the assumptions made in the $K$-matrix method, commonly used in extracting empirically the $\gamma N \rightarrow \Delta$ transition amplitudes from the data, are consistent with