Vertex Poles and Bound States in the Lee Model

We study the $Z=0\mathrm{}$ limit of a version of the Lee model, recently introduced and solved by Bronzan, from the point of view of recent work by Gerstein and Deshpande. It is shown how vertex function and inverse propagator poles develop and behave for small vertex renormalization constant, and their connection with the bound-state limit is studied. It is found that the condition of finite mass renormalization in the $Z_U=0\mathrm{}$ limit can be satisfied in this model and leads to bootstrap-type results.