Vanishing of Proper Vertex Functions as a Condition for Reggeization of Elementary Particles

The connection between Regge poles and elementary-particle poles was explored in a previous paper, where we made use of the approximation of elastic unitarity and of the assumption of no Castillejo-Dalitz-Dyson (CDD) ambiguity. In this paper we include the CDD ambiguity, and show without recourse to any approximation that the vanishing of Γ(s) (the renormalized pion-nucleon proper vertex function with one nucleon off the mass shell) is a necessary and sufficient condition for the elementary nucleon to lie on the Regge trajectory. From the condition Γ(s)≡0 it also follows that Z₁=Z₂=Z₂δM=0, where Z₁ is the vertex-renormalization constant, Z₂ is the nucleon wave-function renormalization constant, and δM is the nucleon self-mass. But the vanishing of Γ(s) does not always follow from Z₁=Z₂=Z₂δM=0. Therefore, the so far widely recognized compositeness condition, Z₁=Z₂=0, or Z₂=0 with finite self-mass, is not sufficient to Reggeize the elementary nucleon. Finally, Kaus and Zachariasen's criterion for Reggeization is discussed. Their criterion is not necessary to Reggeize the elementary nucleon; it should be modified to Γ(s)K(s)≡0 and gB²=0, where gB² is the residue of the extra pole in the pion-nucleon scattering amplitude and $K(s)$ is the pion-nucleon form factor with one nucleon off the mass shell.