High-Spin Baryons. II. Dynamical Mechanisms for Regge Recurrences of Inelastic Resonances; General Theory

Virtual transitions of the type $\pi N_i\to \rho N_j$, where $N_i$ represents an arbitrary nucleon isobar, are considered as the driving force for Regge recurrences of inelastic resonances. We consider only the $s$-wave $\rho N_j$ configuration. Since most of the low-lying isobars have even parity, we are therefore concerned mainly with odd-parity resonances. Regge trajectories of the latter are a consequence of Regge recurrences of the even-parity states. An important example is given by taking $N_i$ to be the sequence: nucleon ${\mathrm{½}}^{+}$ and its recurrence ${\frac{5}{2}}^{+}$ (quantum numbers $P_{11}$ and $F_{15}$ in conventional notation). Isospin and centrifugal-barrier considerations lead directly to the existence of the $D_{13}-G_{17}$ sequence. In this paper the general formalism is set up. Numerical results are given separately. First, however, a simpler derivation is given of the elastic forces due to isobar exchange in the quasistatic approximation. Previous work has shown the relation of these forces to the existence of even-parity Regge trajectories. Then the unitarity relations are set up for helicity amplitudes describing the reactions $N_a\pi \to N_c\pi$, $N_a\pi \to N_c\rho$, and $N_a\rho \to N_c\rho$. Next a general explicit expression is given for the onepion-exchange (OPE) approximation for $N_a\pi \to N_c\rho$. Expressions are given for the above amplitudes in several models, in which only the OPE coupling is considered. The pole approximation is used to solve the many-channel $\frac{N}{D}$ equations. Techniques for handling the general isospin problem are developed and their relevance to the present problem discussed.