Dynamical Test of Constituent Quark Models with $\pi N$ Reactions

A dynamical approach is developed to predict the $\pi N$ scattering amplitudes starting with the constituent quark models. The first step is to apply a variational method to solve the three-quark bound state problem. The resulting wave functions are used to calculate the $N^* \to \pi N, \eta N, \pi\Delta$ vertex functions by assuming that the $\pi$ and $\eta$ mesons couple directly to quarks. These vertex functions and the predicted baryon bare masses then define a Hamiltonian for $\pi N$ reactions. We apply a unitary transformation method to derive from the constructed Hamiltonian a multi-channel and multi-resonance reaction model for predicting the $\pi N$ scattering amplitudes up to $W = 2$ GeV. With the parameters constrained by the $\Delta(1232$) excitation, we have examined the extent to which the $\pi N$ scattering in $S_{11}$ channel can be described by the constituent quark models based on the one-gluon-exchange or one-meson-exchange mechanisms. It is found that the data seem to favor the spin-spin interaction due to one-meson-exchange and the tensor interaction due to one-gluon-exchange. A phenomenological quark-quark potential has been constructed to reproduce the $S_{11}$ amplitude.