Coupled-Channel Amplitudes: Physical Effects and Analytic Structure

An investigation is made of the effects of an indefinite number of coupled channels on two-particle amplitudes. Central coupling, chain coupling, and combinations of them are considered. Some results are general, while others depend on a generalized potential model. The existence of strong resonance-producing effects is related to zeros of the $S$ matrix. Forms of Levinson's theorem are found for both the amplitude and $S$-matrix phases. It is shown that if the effective number of coupled channels increases linearly with the energy that (1) the partial-wave $S$ matrix can vanish asymptotically (the usual result is that $S_L\to 1$); and (2) Regge trajectories rise asymptotically as the square root of the energy. Special results of the MacDowell symmetry are noted for a finite number of coupled channels. With respect to a special (boundary condition) model it is found that Regge trajectories which cross the origin near $L=\frac{1}{2}$ should have much larger slopes than those crossing the origin near $L=1\mathrm{}$, in agreement with observation. Regge cuts result from the coupling of effective three-body channels.