Vertex-Function Poles and the Bootstrap Condition in Field Theory

Abstract

We discuss the implications of maintaining finite mass renormalization when the wave-function renormalization constant $Z_3$ of an elementary particle is set equal to zero, making only the approximations of two-particle unitarity. We show that this defines a field-theory bootstrap for the elementary particle which is completely equivalent to the usual type of bootstrap based on the $\frac{N}{D}$ method. As $Z_3$ goes to zero the vertex function and inverse propagator develop poles which move to ${\mu }^2$, the elementary-particle mass, in the limit. For nonzero $Z_3$ this pole does not contribute to the scattering amplitude, but at $Z_3=0\mathrm{}$ it cancels the elementary-particle pole in the single-particle reducible part, leaving the dynamical pole in the irreducible part. We suggest, further, that in this limit the bootstrapped state is a Regge pole.