Even-wave harmonic-oscillator theory of baryonic states. II. Orbital matrix elements and selection rules

This paper deals with the construction of baryon wave functions in the even-wave harmonic-oscillator (h.o.) model described in a previous paper. This model gives rise to an apparent doubling of (70, $1^{-}$) states and a further splitting of 70 states with higher angular momenta. The split states up to $L=2\mathrm{}$ are constructed in terms of a new set of variables ($x_i,y_i,\lambda$) which are found to be particularly convenient for the description of the orbital symmetries of the 70 states. Evaluation of the orbital matrix elements based on these wave functions leads to a set of orbital selection rules forbidding transitions between the upper ($u$) and lower ($l$) states of (70, $1^{-}$) and similar selection rules for higher states. For the description of the physical processes in this model, the nucleon octet is assumed to be an ideally mixed state of (56, $0^{+}$) and the newly available (70, $0^{+}$) ground state which has no ready counterpart in the full-wave h.o. model. Such a mixed nucleon octet is shown to reconcile at once the $\frac{G_A}{G_V}$ ratio as well as the $\Delta \to N\pi$ width with the $\mathrm{NN}\pi$ coupling constant.