Composite Particles in Separable-Potential Models

Composite particles (bound states) are investigated in three separable potential models---(1) a model with one kind of static heavy neutral scalar boson interacting with one kind of light neutral scalar boson, (2) a model with one kind of static heavy boson and two (or any number of) kinds of light bosons, (3) a model with two (or any number of) kinds of static heavy bosons and one kind of light boson. These models are solved exactly, the last one only in the case of equal interaction form factors. The $S$-matrix elements are calculated both directly and dispersion-theoretically. The allowed physical processes are found to be elastic scatterings only; composite particles can exist but can neither be created, nor destroyed, nor altered in any way within the framework of the models studied. The $S$-matrix elements are shown to be independent of whether the scattering target is a number of heavy bosons or any composite particle which could be formed by the same number (and kinds) of heavy bosons. The composite particle states are identified as those eigenstates of the Hamiltonian for which the total number of heavy bosons is nonzero, the number of free heavy bosons is zero, and the total number of light bosons is greater than the number of free light bosons. The energy of such a state is less than the rest energy of its components, i.e., the state is a bound state. Particle operators for the composite particles are constructed but are observed to lack satisfactory Lehmann-Symanzik-Zimmermann asymptotic limits.