Angular momentum projection with quantum effects.

Abstract

We have improved a simple and rapid method of calculating expectation values of operators in states of good angular momentum projected from a hedgehog baryon state introduced by Birse et al. We have included the contributions of quantum mesons, while in the original method only classical meson fields were included. The method has been applied to models where the mean-field approximation does not include loop terms. Hence, for reasons of consistency, contributions of quantum loops to the matrix elements have been dropped. The symmetry of the hedgehog state under grand reversal (the combined operation of time reversal and ${\mathit{e}}_{2}^{\mathit{i}\mathrm{\ensuremath{\pi}}\mathbf{I}\mathrm{^}}$, where I^ is the isospin operator) introduces remarkable simplification in the calculation of matrix elements of operators which do not contain time derivatives of meson fields. The quantum meson contributions turn out to be 3/2/〈B\ensuremath{\Vert}J${\mathrm{^}}^{2}$\ensuremath{\Vert}B〉 times the classical meson-field contributions, with \ensuremath{\Vert}B〉 being the hedgehog state. Such operators are encountered in the calculation of nucleon magnetic moments, ${\mathit{g}}_{\mathit{A}}$(0) and ${\mathit{g}}_{\mathrm{\ensuremath{\pi}}\mathit{N}\mathit{N}}$(0)/2M. Calculation of charge radii involves operators containing time derivatives of meson fields and requires the knowledge of wave functions of quantum mesons. Proper nonperturbative treatment, even though at the tree level, requires that these wave functions describe the motion of the mesons in the potential generated by the baryon. Fortunately, because of the neglect of the loop terms, one needs only the even-parity, grand-spin-1 states which are purely pionic. The Goldberger-Treiman relations, an exact result for the model, serves as a partial test of the method of calculation discussed here. This has been used to demonstrate the remarkable improvement in the results produced by the inclusion of quantum effects of the mesons.