Dispersive Treatment of Weak Decays and Final-State Interactions in Model Theories

A model, in which a heavy fermion $B$ is added to the Lee model and weakly coupled to $V$ and $\theta$ is considered. Decay amplitudes for $B\to V+\theta$ and $B\to N+\theta +\theta$ are evaluated by dispersion theoretic methods. The absorptive part of these amplitudes incorporate contributions from one- and two-boson intermediate states. Attention is focused on the question of how well founded is the usual approximate treatment of absorptive amplitudes, which neglects the higher mass states (here the two-boson states) with respect to the lowest mass (here the one-boson) states. It is shown that in a dispersion scheme which involves sufficient subtraction to give completely finite results, and which involves as many arbitrary constants as the theory allows, the one-particle contributions to the absorptive parts of the $B\to N+\theta +{\theta }^{\prime }$ amplitude diverge logarithmically and that these logarithmic divergences are cancelled by the two-particle contributions to this amplitude.