Dynamical test of constituent quark models with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>N</mml:mi></mml:math>reactions

Abstract

A dynamical approach is developed to predict the $\ensuremath{\pi}N$ 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 ${N}^{*}\ensuremath{\rightarrow}\ensuremath{\pi}N,\ensuremath{\eta}N,\ensuremath{\pi}\ensuremath{\Delta}$ vertex functions by assuming that the $\ensuremath{\pi}$ and $\ensuremath{\eta}$ mesons couple directly to quarks. These vertex functions and the predicted baryon bare masses then define a Hamiltonian for $\ensuremath{\pi}N$ reactions. We apply a unitary transformation method to derive from the constructed Hamiltonian a multichannel and multiresonance reaction model for predicting the $\ensuremath{\pi}N$ scattering amplitudes up to $W=2$ GeV. With the parameters constrained by the $\ensuremath{\Delta}(1232)$ excitation, we have examined the extent to which the $\ensuremath{\pi}N$ scattering in the ${S}_{11}$ channel can be described by constituent quark models based on 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.