Absorptive Processes and Single-Particle Exchange Models at High Energies. I. General Theory

Abstract

It has been shown by several authors that the effect of competition from other open channels can materially affect the predictions of single-particle exchange models for high-energy particle reactions. The general theory of these absorptive effects in the distorted-wave Born approximation is discussed in detail for several situations in nonrelativistic potential scattering using the WKB approximation. The case of many coupled two-body channels is considered in detail. If the transition potential in question is of short range compared with most of the other potentials, the distorted-wave Born approximation for the $S$ matrix in the presence of this potential is given by $S=S_0+A^T{S}^{B}A$, where $S_0$ is the $S$ matrix in the absence of the perturbing potential $S^B$ is the Born approximation for the transition $S$ matrix, and $A$ is a known unitary matrix such that $S_0=A^{T}A$. Under appropriate circumstances, $A$ may be approximated by the square root of the $S$ matrix; one then obtains the generalization to the many-channel problem of a result derived previously for two channels, $S=S_0+{S_0}^{\frac{1}{2}}{S}^B{S_0}^{\frac{1}{2}}$. For long-range transition potentials, $S$ is given by $S=S_0+\frac{1}{2}[S_0{S}^B+S^B{S}_0]$. These results do not depend on the details of the potentials. It is therefore plausible that they may be used in relativistic calculations. The generalizations necessary for the application to relativistic particle reactions, including the effects of the particle spins, are discussed in detail. The properties of the rotation coefficients of the second kind encountered in this analysis are discussed in an Appendix. A second Appendix deals with the general Fourier-Bessel integral representation of the scattering amplitudes for particles with spin. Finally, the possibility of adapting our results to a $K$-matrix formalism, appropriate when the distorted-wave Born approximation fails, is discussed briefly.