Bosonization in the presence of confinement: Calculation of the nucleon-nucleon interaction.

Abstract

We describe an extended version of the Nambu\char21{}Jona-Lasinio (NJL) model that includes a description of confinement. It is necessary to incorporate some description of confinement in order to discuss the properties of the sigma, rho, and omega mesons in the NJL model. These mesons, in addition to the pion, are the minimum needed to describe the salient features of the nucleon-nucleon interaction. In previous work we considered the relation between the bosonized NJL model and the one-boson-exchange (OBE) model of the nucleon-nucleon force. Most of our attention was given to pion and sigma exchange. We provide a review of that work and extend our discussion to a consideration of rho and omega exchange. We also present a more detailed discussion of the bosonization procedure. Our results depend upon the strength of the confining interaction. Once that is fixed, we obtain good values for the omega-nucleon coupling constant, ${\mathit{G}}_{\mathrm{\ensuremath{\omega}}\mathit{NN}}$, and for the tensor coupling constant ${\mathit{f}}_{\mathrm{\ensuremath{\rho}}}$, in the rho-nucleon interaction. (One limitation of the present version of the model is that the ratio ${\mathit{f}}_{\mathrm{\ensuremath{\rho}}}$/${\mathit{g}}_{\mathrm{\ensuremath{\rho}}}$=3.70, instead of the empirical value of ${\mathit{f}}_{\mathrm{\ensuremath{\rho}}}$/${\mathit{g}}_{\mathrm{\ensuremath{\rho}}}$\ensuremath{\simeq}6.1.) If we consider nucleon-nucleon scattering for relatively small momentum transfer, we obtain good results for the processes of sigma, pion, rho, and omega exchange. Remarkably, the description of pion exchange is very accurate up to ${\mathit{q}}^{2}$\ensuremath{\sim}-2 ${\mathrm{GeV}}^{2}$. That is, the microscopic model reproduces the pion-exchange amplitude of the boson-exchange model over a broad range of momentum transfer when we specify a single parameter than governs the momentum-transfer dependence of the pseudoscalar-isovector form factor of the nucleon. In the other channels (\ensuremath{\sigma},\ensuremath{\rho},\ensuremath{\omega}), the nucleon form factors may be treated in the same manner. However, if we calculate the form factors in our model, we find that they are too ``soft'' to fit the OBE amplitudes away from ${\mathit{q}}^{2}$\ensuremath{\simeq}0. Further work is needed to obtain good fits for the various amplitudes for large momentum transfer, although the OBE amplitudes are well reproduced in the case of scattering at small momentum transfer (\ensuremath{\Vert}${\mathit{q}}^{2}$\ensuremath{\Vert}\ensuremath{\le}0.1 ${\mathrm{GeV}}^{2}$). \textcopyright{} 1996 The American Physical Society.