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

Abstract

Virtual transitions of the type $\ensuremath{\pi}{N}_{i}\ensuremath{\rightarrow}\ensuremath{\rho}{N}_{j}$, 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 $\ensuremath{\rho}{N}_{j}$ 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 ${\mathrm{\textonehalf{}}}^{+}$ 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 ${D}_{13}\ensuremath{-}{G}_{17}$ 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 ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\pi}$, ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$, and ${N}_{a}\ensuremath{\rho}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$. Next a general explicit expression is given for the onepion-exchange (OPE) approximation for ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$. 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.