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 $\Gamma \mathrm{}\left(s\right)\mathrm{}$ (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 $\Gamma \mathrm{}\left(s\right)\mathrm{}\equiv 0$ it also follows that $Z_1=Z_2=Z_2\delta M=0\mathrm{}$, where $Z_1$ is the vertex-renormalization constant, $Z_2$ is the nucleon wave-function renormalization constant, and $\delta M$ is the nucleon self-mass. But the vanishing of $\Gamma \mathrm{}\left(s\right)\mathrm{}$ does not always follow from $Z_1=Z_2=Z_2\delta M=0\mathrm{}$. Therefore, the so far widely recognized compositeness condition, $Z_1=Z_2=0\mathrm{}$, or $Z_2=0\mathrm{}$ 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 $\Gamma \mathrm{}\left(s\right)K\left(s\right)\mathrm{}\equiv 0$ and ${g_B}^2=0\mathrm{}$, where ${g_B}^2$ is the residue of the extra pole in the pion-nucleon scattering amplitude and $K\left(s\right)\mathrm{}$ is the pion-nucleon form factor with one nucleon off the mass shell.